
s&w?& 



FINANCE 

AND 

LIFE INSURANCE 



A Hand Book of Tables and 
Formulae, with Rules and 
Explanations, for the use of 
Lawyers, Brokers, Bankers 
Insurance Men and Others 



By 
WILLIAM A. DUDLEY 

ATTORNEY-AT-LAW 



COPYRIGHT, 1916, 

BY 

WM. A. DUDLEY. 






17 1916 



©CI.A446460 

! . 






PREFACE 

The object had in view in the preparation of this book has been 
to place in the hands of my professional brethren an instrument of 
which I, early in my practice, felt the urgent need, that is, the data 
and information necessary to readily advise clients as to the value 
of securities, incomes, insurances, estates and other instruments of 
finance and investment. The tendency is stronger now than ever for 
lawyers to serve in the capacity of business experts and in every office, 
there is frequent occasion to pass upon the value and soundness of 
some investment, security or estate. The forty-three tables and 
explanatory text matter contained in the succeeding pages, it is believed, 
afford all of the facts and information necessary in order to answer 
most of the practical questions that arise on the face of the instruments 
of finance and life insurance. A second object sought has been to 
provide the banker, broker and financier an easy means of making 
the calculations called for in the conduct of his business and in advising 
his customers. And finally, it is believed that the practical insurance 
man will find in these pages much that will be helpful. No effort has 
been made to develop the scientific principles involved in finance and 
insurance. That has been left to the actuaries whose business it is 
to teach those subjects, but the endeavor has been to apply the prin- 
ciples and rules demonstrated by the actuaries to the solution of the 
practical problems arising in these great and ever widening fields. The 
purpose has been to make the subjects simple and the scientific principles 
have only been discussed when the attainment of tins object seemed to 
require it. The writer takes this occasion to acknowledge his obli- 
gations to the Acturial Society of America for permission to republish 
its regraduated American Experience table of mortality and force of 
mortality, and publish annuity tables derived therefrom; to the 
Connecticut Mutual Life Insurance Company, for permission to re- 
publish its three per cent American Experience Commutation columns 
and annuities, and to Mr. Abb Landis, actuary, for permission to 
republish the National Fraternal Congress table, commutation tables 
and annuities computed by him. The preparation of the materials 
and work of reducing them to the form here presented has covered a 
period of several years, during which time, the writer has been actively 
engaged in the practice of law, and it would be remarkable indeed if 
no errors should be f ound ; but it is believed that both the rules and the 
tables given may be safely employed for the purposes for which they 
are intended, and it is hoped that they may be found useful. 
Troy, Mo., June, 1916. 



CHAPTER I 
Of Short Methods 

Facility in computation is a thing to be acquired by practice 
intelligently followed, and not many of the so-called short methods 
possess any practical value. Grouping in addition, like phrasing in 
stenography, gives speed, but this will come from intelligently directed 
practice. In fact, an active computer soon finds himself employing 
combinations of numbers which greatly facilitates his work but rules 
would have been of little use to him in forming them or in employing 
them in practice. In nearly all of the operations involved in the matters 
treated of in this book, decimal fractions are employed and in many of 
these, figures must be discarded beyond a limited number of decimal 
places. Usually, not more than six decimal places are retained. What 
has long been known as the contracted methods of multiplication 
and of division have been found helpful and will be noticed here. 

1. Multiplication of decimals by the contracted method consists 
in disregarding all figures in the product beyond the number of decimal 
places intended to be retained except that they are considered in car- 
rying to the right hand figures of the partial products. To use this 
method, first determine from the nature of the factors how many 
integral figures will occur in the product, then include as many decimals 
as will be required, dropping those falling to the right. 

Let it be required to multiply 4.549383 by 2.854339 correct to 
six places of decimals. We know there will be two whole numbers 
and six decimals in the product and eight places will be required. 
The multiplication begins with the left hand figure of the multiplier 
and it is generally more convenient to reverse the multiplier. In this 
case, they appear as follows: 

4.549383 
9 334582 



9 098766 


3 639506 


227469 


18198 


1365 


136 


41 


12.985481 



First, the entire multiplicand is multiplied by 2. Next, we begin 
by multiplying 3 by 8, but the product is not entered, then, we have 
8X8, to which we add 2 from the product of 8X3, put down the 
right figure under 2 and carry 6 as in ordinary multiplication. The 



SHORT METHODS OF COMPUTATION. 5 

only other thing to be noticed is that the first and sometimes the 
second figure to the right of the figure above the partial multiplier 
is to be multiplied in order to determine how many to carry. Thus 
we say 5X8=40 and carry 4, then 5X3=15, and add 4, putting 
down 9 in the first column. Next, 4x8=32, then 4x3+3 = 15, and 
finally, 4x9+2=38, and we put 8 in the first column. The rule 
being, in carrying tens from the nearest rejected figure of the multi- 
plicand, to carry 1 additional for any number in units place over 
five. 

2. The contracted method of division of decimals also saves 
time and space and should be employed. 

It consists in cutting off those figures extending beyond the number 
of places intended to be retained, carrying into the partial products 
the proper numbers to avoid loss by the omission of figures from the 
right of the divisor in multiplying. Thus, let it be required to divide 
11.46992 by 36.78559, correct to six places of decimals. We know 
36 will go into 11.4, .3 times and hence there will be no integers in 
the quotient. Arrange the dividend and divisor in the usual way and 
divide: 

36.78559 )11 . 46992 ( . 311533 
11 03568 
43424 
36786 



5638 

3679 

1959 

1839 

120 

110 

10 

11 



In dividing, we reject the figure nine but carry 3 into the product 
of 3 X5, putting down 8. For the next divisor, we reject 5 and find 1 
as the quotient figure then to 1x5 we add 1 for the rejected 59 and 
put down 6 in the partial product, the rule for carrying being the same 
as in contracted multiplication. 

3. But the most valuable instrument yet devised for shortening 
the processes of multiplication, division, involution and evolution is 
the table of logarithms. It not being the object of this book to teach 
mathematics, but rather to apply the mathematics in the solution of 
problems in finance, and life insurance, a discussion of the principles 
of the subject will not be undertaken. But such an explanation of 
the tables and the methods of using them will be given as will enable 
the reader to apply them in the solution of problems. 



CHAPTER II 

Of Logarithms 

1. Logarithms are exponents of the powers of some number 
which is taken as a base. In the tables of logarithms given in this 
book, the number 10 is taken as the base and all numbers are con- 
sidered as powers of ten. And, since 

10° = 1 the logarithm of 1 is 
lO^lO the logarithm of 10 is 1 

10 2 = 100 the logarithm of 100 is 2 

10 3 = 1000 the logarithm of 1000 is 3 

and so on, the logarithm of any number between 1 and 10 is between 
and 1, and hence, is a fraction; the logarithm of any number between 
10 and 100 is between 1 and 2 and hence is 1 plus a fraction; and the 
logarithm of any number between 100 and 1000 is between 2 and 3 
and hence is 2 plus a fraction and so on. 

Again, in the case of fractional numbers, if we take the logarithm 
of the denominators, we have the same rule applying except that the 
sign, by the theory of exponents, becomes negative when the denomina- 
tor is transferred to the numerator, thus: 

11 11 
= =— 1, = =—2, and so on, 



Log. 10 10 1 Log. 100 10 2 

so that the logarithm of a number between 1 and . 1 is between and 
— 1 plus a fraction; the logarithm of any number between .1 and .01 
is between —1 and —2 and hence is —2 plus a fraction. The integral 
part of the logarithm, whether positive or negative, is called the 
characteristic and the fractional part which is always positive is 
called the Mantissa. 

2. The determination of the characteristic of a logarithm will be 
better understood by considering the following exhibit: 

The number 7436 is between the third and fourth powers of ten and 
hence is 3 p i us a fraction. In fact, 7436 = 10 3 - 87134 . Hence, the logar- 
ithm of 7436 is 3 . 87134. Also 743 . 6 is between the second and third 
powers of 10, hence is 2 plus a fraction. In fact, log. 743 . 6 is 10 2 ' 87134 . 

We may arrange the following scheme : 

103.87134 = 7436 or log> 7436=3.87134 

102.87134 = 743 6 or log# 743.6=2.87134 

10 i.87U4 = 74 36 or log- 74.36 = 1.87134 

10 o.87i34 _ 7 436 or log. 7.436 = _. 87134 

10-i-87i34 = .7436 or log. .7436 = 1.87134 

10-2-87134 = 07436 or log. .07436=2.87134 
and so on. 



LOGARITHMS— THE CHARACTERISTIC. 7 

3. From the foregoing, we may derive the rule for finding the 
characteristic of a logarithm. 

The characteristic of an integral or mixed number is one less than 
the number of integral places. When the number is entirely decimal, 
the characteristic is negative and numerically one greater than the 
number of O's between the decimal point and first significant figure. 

4. In practice, two methods are used in writing the characteristic 
of a decimal number. These may be understood from the following 
illustration: 

Log. 7.436 =10.87134-10 or _. 87134 

Log. .7436 = 9.87134-10 or 1.87134 

Log. .07436= 8.87134-10 or 2.87134 

Log. .007436= 7.87134-10 or 3.87134 

and so on, the minus sign over the characteristic indicating that it is 

negative. 

5. It will be noticed from the above illustration that the same 
sequence of figures has the same mantissa and that the characteristic 
alone is changed by moving the decimal point ; that is, by multiplying 
or dividing the number by ten. 

6. It was first purposed by the writer to arrange and publish a 
table of common logarithms to six places, but it was concluded that 
the space it would occupy would be out of proportion to the benefit 
it would confer in view of the fact that such tables are generally avail- 
able. There is included, however, a table which will meet most of 
the requirements of the work and occupies little space. The first 
part, Table la, may be used to find the logarithms of numbers up to 
say, four or five significant figures, by interpolating, true to five decimal 
places by using the first five left hand figures of the group. Thus, the 
mantissa of log of 264 would be found in column 4 opposite 2 . 6 under 
N. and to five places is .42160. The mantissa of 269 would be found 
in column 9 opposite 26 and to five places is .42975. The character- 
istic according to principles explained in the preceding sections is in 
each case 2 and the logs are, respectively, 2.42160 and 2.42975. Had 
the problem required the log of 26472, we would first have to find the 
log of 264 as above and add to it seventy-two hundredths of the dif- 
ference between the logs of 264 and 265. We find log of 265 on the 
same line in column 5. It is .42325 and subtracting .42160, we have 
for the difference, .00165. Multiplying the latter by .72, we have 
.00118.80. We call it .00119 and add it to .42160. This with the 
characteristic 4, gives 4 . 42279 as the logarithm. By the seven figure 
table, the log of the same number is given as 4.4227868 so that the 
result is correct to five places, which is sufficient for most purposes. 
The tables published may also be used to obtain the logarithms of 
any number to eight places, by methods which will be explained 
further on. 

7. From the last section we may, for finding the mantissa, derive 
the following: 



8 FINANCE AND LIFE INSURANCE. 

Rule 

For a number consisting of three figures the logarithm will be found 
in the column headed with the right hand figure of the number and in 
the line opposite the two left hand figures of the number where they 
appear in the left hand column of the table under the letter N. To 
find the logarithm of a number consisting of four or more significant 
figures, first find the log of the three left hand figures in the manner 
just described. Then multiply the difference between the log thus 
found and the log of the number in the next column at the right, by the 
remaining figures of the number at the right and add the product as 
a correction to the original logarithm found. This will give the man- 
tissa. For the characteristic of the number, if the number is integral, 
or partly integral, write for the characteristic a number one less than 
the number of whole numbers in the given number. If the number 
is a decimal with one or more ciphers between the decimal point and 
the significant figures, write the characteristic negative and for one 
cipher write 0, for two ciphers 1, for three ciphers 2, and so on. Thus 
the logarithm of .0004724 is 2.67431, and of .000004724 is 4.67431 
(see Section 4). 

8. To find the number corresponding to a given logarithm, often 
called the anti log, we suggest the following: Neglecting the charac- 
teristic for the time being, find, if possible, the logarithm in the table. 
The two left hand figures of the number will be found in the left hand 
column of the table under N and opposite the logarithm found and the 
units figure will be found at the top of the column in which the log was 
found. Thus the number corresponding to the log 2.68395 is 483. 
If the exact mantissa can not be found in the table, find the one less 
and nearest to the given log. The three left hand figures will be found 
as already explained. The next figure, or figures, at the right of the 
number will be the quotient found by dividing the difference between 
the given log and the log first found by the difference between the 
first log found and the log of a number one unit greater. Thus take 
the logarithm 2.57702. The nearest mantissa in the table is .57634 
found under 7 opposite 37. The last three figures of the number 
sought are therefore, 377. The difference between the given log and 
the log of 377 is .00068. The difference between the log of 377 and 
the log of 378 (57749) is . 00115 and 68^- 115 = . 591. The right hand 
figure being less than five is rejected and 59 affixed to the original 
number and we have 37759. The characteristic being 2, three of the 
figures are integral and two decimal and the number is 377 . 59. The 
table la may be used with confidence to five places as a rule, but some- 
times the quotient should be carried one or two places further so that 
if the rejected figure is greater than five, the last retained figure must 
be increased by one. If less than five, the rejected figures are disre- 
garded. 

9. We know by the theory of exponents that the exponent of 
the product of two or more numbers may be expressed as the sum of 



MULTIPLICATION, DIVISION, POWERS AND ROOTS. 9 

the exponents of the numbers, hence for multiplication, we have 
the Rule: 

The sum of the logarithms of two or more numbers is the logarithm 
of their product. The anti log of this sum is the product of the num- 
bers. Thus to find the product of 15 . 13 by 1 .348, we take the logs: 

Log of 15.13 =1.17984 
Log of 1.348 = .13001 



Their sum is 1.30985 =20.4104, answer. 
10. For division, we have the Rule: 

Subtract the logarithm of the divisor from the logarithm of the 
dividend, the remainder will be the logarithm of the quotient. The 
anti log or corresponding natural number will be the quotient. 

Thus to divide 3 .390 by 84.51, we have 

Log 3 . 390 = . 53020 = 10 . 53020 - 10 
Log 84.51=1.92691= 1.92691 



8 . 60329 - 10 = . 0401 14, answer. 

11. Since a common fraction is merely an indicated division, the 
logarithm of a fraction is the difference found by subtracting the 
log of the denominator from the log of the numerator. Thus the log 
of 3 /*=log 3 -Log 4. 

12. The logarithm of a power of any number may be found by 
multiplying the log of the number by the exponent of the power to 
which the number is to be raised. 

13. As an example, let it be required to raise 1.065 to the 20th 
power. By the rule (Art. 12) (1.065) 20 =20 log 1.065. Now, log 
1 . 065 = . 02735 and 20 X . 02735 = . 54700 = 3 . 5237. 

14. The logarithm of the root of a given number is the quotient 
found by dividing its logarithm by the index of the root. 

15. As an example, let it be required to find the cube root of 625. 
By the rule (Art. 14) log 625=2.79588. The latter sum divided 

by three = .93196 =8 . 55, Answer. 

16. The co logarithm of a number is found by subtracting the 
given logarithm from the log of 1, that is, from 0, written as 10 — 10. 
To add the co log is equivalent to deducting the log of the given 
number, that is, to dividing by that number. 

17. The principle of Article 12 enables us to find readily the 
value of an exponential equation, that is, an equation in which the 
exponent of one of the numbers is an unknown quantity. 

As an example, let it be assumed that the amount of a loan for a 
certain time at 5 per cent, is 2 .65330. What is the term, the principal 
being 1? 



10 FINANCE AND LIFE INSURANCE. 

Here, (1.05)* =2.6533 and x log 1.05 =log 2.6533. 

And finally x = log 2.6533-Hog 1.05 = .42379-^ .02119 .=20. 

Answer, 20 years. 

18. Tables la and lb may be used to find the logarithms and 
anti logs of numbers to eight places. Their use is based upon the 
following principles, already noticed, viz., the mantissa of the log 
of a number depends on the sequence of the figures and not upon the 
location of the decimal point. The logarithm of a number is equal 
to the sum of the logs of its factors. Also the log of any number 
less than 10 is wholly decimal and may be resolved into two factors, 
one of which will be between 9 . 99 and 1 . 00, and the other between 
1 . 10 and 1 . 0099900. Table la gives the first set of factors mentioned 
and Table lb the second set, carried to eight decimal places. 

19. As illustrating the use of these factors to find the log of a num- 
ber, let it be required to find the logarithm of the number 56947848 
true to eight places. We may suppose for the purposes of the solu- 
tion that the decimal point is placed after 5 and we may then find 
the log of 5 . 69 in part la of the table. It is on the last page of part 
la and is .75511227. Now dividing 5.6947848 by 5.69, we obtain 
10008409 as the other factor. Its logarithm appears on page 1 of 
Table lb, and is .00036465 for the number 1.00084. We require 
then the log of .0000009. The difference between 6465 under 4 and 
6899 under 5 is 434, which multiplied by .09, gives .039 as the cor- 
rection to the log of 1 .00084. We now have as the log of both factors, 
the following values: 

Of 5.69 =.75511227 

Of 100084 =00036465 

Correction to latter 039 



Total .75547731, which is the mantissa. 

The characteristic on principles already explained is 7, so that the 
required log is 7.75547731. 

20. To find the natural number corresponding to a given logar- 
ithm, we in effect, reverse the process just employed. Thus, we first 
find in Table la the largest logarithm that can be taken from the 
given log. This is the log of one factor. Deduct it from the given 
log and the remainder is the other factor. We then "take out" the 
corresponding natural numbers. These are the factors. Their 
product is the number. 

As an example, let it be required to find the number corresponding 
to log 4.48368759. Disregarding the characteristic for the moment, 
and looking into table la we find . 48287358, the anti log of which is 
3 . 04. Then 48368759-48287358 = . 00081401. Looking into Table 
lb, we find as the nearest log to the latter sum, .00081137 which is 
the log of 1.00187. But 137 from 401 gives .00000264 as the dif- 



LOGARITHMS BY FACTORS. 11 

ference between the factor log and the nearest log thereto. The dif- 
ference between the logs of 1 .00187 and 1 .00188 is .00000434, dividing 
264 by 434 gives 61 which is affixed to the number found and we have 
as the second factor 1.0018761, multiplying this by the other factor, 
304, we have 30457033. Recurring now to the characteristic, we 
know that there must be five figures in the integral part so that the 
number sought is 30457 . 033. 

21. From the foregoing, it is believed a working knowledge of 
logarithms may be realized without much difficulty by those not already 
familiar with their use, though the treatment of the subject has neces- 
sarily been very imperfect indeed. The use of logs is so helpful, and 
sometimes necessary, in the computation of the values dealt with 
thruout the work, that it was deemed proper to devote a little space 
to the subject. 



CHAPTER III 

Of Series 

In the solution of many of the problems of finance and insurance, 
a knowledge of the method of developing a function into a series is 
helpful or necessary, but only a few of the more important principles 
can be noticed here. 

1. An infinite series may be developed in some cases by division, 
by the principle of indeterminate co-efficients or by the bi-nominal 
theorem. Taylor's and MacLauren's theorems are also available, 
the former in particular being very general in its applicability to the 
expansion of functions. 

2. The method of finite differences may be employed to extend a 
series or to insert intermediate terms and also to compute the sum of 
a series, where a sufficient number of terms are given to represent 
the law governing the development of the series. The subject will 
be briefly noticed in the following Articles: 

3. Let it be required to extend the series 1, 8, 27, 64, 125, to 
8 terms. 

Arrange the given terms in a column and difference them indicating 
first, second, and so forth differences by the symbols, A, A 2 , A 3 , 
and so on. We then have the following scheme : 

Terms Series Differences 
A A 2 A 3 

1 1+ 7+12+6+Q 

2 8+ 19 + 18 6 

3 27+ 37+ 24|+6 

4 04 + 61-h ) 30 

5 125+91+36 

6 216 + 127+42 

7 343+169 

8 512 

In the process, we first deducted each of the numbers from the 
one below it forming the first order of differences, 7, 19, 37 &c. Then 
each of these was deducted from the one below it forming the second 
order of differences. The numbers 7, 12, and 6 are called the leading 
differences and the entire series may be formed from these. Here 
we have the series to the 5th term. We add 6 to 24 putting 30 in the 
second order, then add 30 to 61, putting 91 in the first order, and then 
add 91 to the series, making the 6th term. The others were obtained 
in the same manner. The third order of differences consists of 6 and 
6, that is, are constant, giving as the 4th leading difference. Such 



DEVELOPING SERIES— NTH TEEM. 13 

a series may be summed exactly by the methods of finite differences. 
And also one in which one of the higher orders of differences may be 
assumed to be constant without material error may be summed ap- 
proximately. 

4. Suppose it were desired to find the 15th term of the series of 
the last article directly. In such case, we may use the formula: 

(n-1) (n-2) (n-1) (n-2) (n-3) 

U n = a + (n-l)A + A 2 + A 3 +etc. 

1.2 1.2.3 

in which a is used for the first term, 1. n =15, the number of terms, 
and, as we have seen A =7, A 2 = 12 and A 3 =6. The 4th difference 
we have seen is 0. Inserting these values in the formula, we have 

1+14 7 +14 13 12 14 13 12 

+ 6 = 1+98 + 1092+2184=3375. 

2 2 3 

The foregoing formula is called the formula for the nth term, 
and is an important one, and should become familiar to the reader. 

5. A method which is adapted to the development of a series of 
terms which substantially follow some law and which do not change 
rapidly is to obtain the values of a number of equidistant terms, such 
as the 5th, 10th, 15th and so forth, or the 5th, 15th, 25th, 35th and 
etc., then assume. 

ux=A+Bx+Cx 2 +Dx 3 +etc. Then, if the intervals be five, 
that is, x =5, we have: 

u-A. 

u5 = A +5B +25C +125D 
ulO = A + 10B +100C + 1000D 
ul5 = A + 15B +225C +3375D 

By inserting the given values of uO, u5 and etc. in the several 
equations, we may compute the values of the co-efficients, A, B, C 
and etc. by differencing. 

6. An instructive example of this method is given, in the Institute 
of Actuaries Text Book at page 435, illustrating the construction of a 
mortality table. 

Having given li =2844 l 3 o=2501 

1 20 =2705 1 40 =2236 to 

find the intermediate values of 1. Here we may assume bo ^Uo, 
I20 =Uio, I30 =U2o and I40 =u 3 o and the formula appears as follows: 

2844 =u =A 

2705 =uw = A + 10B + 100C +1000D 
2501 =u 20 =A+20B+400C+8000D 
2236 =u 30 = A +30B +900C +27000D 



14 FINANCE AND LIFE INSURANCE. 

Differencing both sides of the equations we have: 

- 139 = 10B -flOOC + 1000D 
-204 = 10B +300C +7000D 
-265 = 10B +500C + 19000D 

Differencing these equations we have: 

-65=200C+6000D 
-61=200C+12000D 

Differencing these equations we have: 
4=6000D 

D= 4+ 6000 = .0006 
-65=200C+4 

C = -694-200= -.3450 
-139 = 10B+-34.50 + .6 

B= -105.1664-10= --10.5166 
A =2844 

In =A+B+C+D=2844 + ( -10. 5166) +(-.3450) + .0006 =2833 
1 12 =A +2B +4C+8D = 2844 + ( -21 .0333) + ( -1.3800) + .0053 =2822 
lis =A+3B+9C +27D = 2844 + ( -31.550) +(3.1050) +.0180 =2809 

The method of procedure is sufficiently disclosed to enable the reader 
to continue the series. 

7. The method of Article 4 may be used to insert terms in a series 
by taking n fractional and finding the nth term in the manner illus- 
trated in that article. The process is called interpolation. 

8. As an example, let it be required to find the value of an annuity 
at 4 x / 4 % computed by the American Experience table of mortality, 
the age of the nominee being 30 years, having given the values at 
3, 372, 4 and 472 per cent, respectively, as follows: 

20.093, 18.605, 17.291 and 16.124. 

Here n = 7 / 2 and we arrange the functions and difference as follows: 



ate 


Annuity 


Differences 








A 


A 2 


A 3 


3 


20.093- 


-1.488 + 


.174-. 


027 


37* 


18.605- 


1.314+147 




4 


17.291- 


-1.167 






4V 2 


16.124 









Inserting the values found in the formula, we have : 

A30=20.093+ 5 /2-(-1.488)+ 5 / 2 -72- 1 /2-174 + 5 /2- 3 / 2 - 1 /2 
20. 093 -3. 720 +.326 -.008 = 16,691, Answer. 

9. The last method is available only where the given terms repre- 
sent equidistant values of the function. The distance between the 
rates in the given case is 7 2 % and is counted as 1 in finding n. The 
nth term on this theory would be 3 + Va = 7 A and (n — 1) = h l 2 . 



METHODS OF INTERPPOLATION. 15 

10. The formula for finding the nth term, Article 4, may some- 
times be used where only a few consecutive terms and a distant term 
are given, to find intermediate terms. Thus, to take another example 
from the Text Book, we have: 
u =96779, ux =97245, u 2 =97624, u 9 = 100000, to 
find the terms from u 3 to u 8 , inclusive. 

Here we have three equidistant terms from which first and second 
differences may be obtained, thus: 

A A 2 
u =96779+466 -87 
ui =97245+379 
u 2 =97624 

The third leading difference is required. By the formula 

9-8 9-8-7 

u 9 =u +9Au A 2 u +- A 3 u =When simplified u +9Au + 

1-2 1-2-3 

36A 2 u +84A 3 u 8 . Transposing terms, we have 
-84 A 3 =u +9 A +36 A 2 -u 9 . 
Dividing through by 84 

u 9 - (u +9 Au +36 A 2 u 100000 - (96779 +4194-3132) 

A 3 = = =25.7 

84 84 

In the problem u 9 is the 10th term. 

Proceeding now, as in Article 4, we have three leading differences: 

A A 2 A 3 

466,-87 and 25.7 and the first term 96779 from which to find the 
others. For the 9th term, u 8 , we have: 

8-7-(-87) 8-7-6-25.7 

96779 +8 • 466 + + = 

12 123 

96779 +3728 - 2436 + 1439 . 2 = 99510. 

For u;, we have 96779 +7 466 + 7 6 ( -87) 



2 
7 6 5 25.7 

=96779+3262-1827+899=99113 

6 

For u 6 , we have 96779+2796-1305+514=98784. 
For u 3 , we have 96779 + . 466 3 +3 • 2 • ( - 87) 

+ 

2 
3 2 1 25.7 

— =97942. And so on, and thus we may find the first ten 

12 3 



16 FINANCE AND LIFE INSURANCE. 

values of the mortality table as follows: 

lio = 100000 1 15 =98224 

l u = 99510 1 16 =97942 

1 12 = 99113 1 17 =97624 

li3= 98784 lis =97245 

1 14 = 98496 1 19 =96779 

11. Lagrange's Theorem, may be conveniently employed 
where only one or two terms are required and the data to be used 
consists. of terms which are not equidistant. The theorem is some- 
times expressed as follows : 

u x =A(x— b) (x — c) (x — d) .. . . (x — n) 
-fB(x-a) (x-c) (x-d) . . . (x-n) 
-f-C(x— a) (x— b) (x— d) . . . (x— n) and so on to n terms. 

The values of A, B, C and etc. are determined by substituting a, b, c 
and etc. respectively in the successive members, and simplifying the 
result, by which means the formula is expressed as follows : 

u x u a 



(x-a) (x-b) (x-c) . . . (x-n) (x- a) (a -b) (a-c) (a- d) . . (a-n) 

u b 

+ 



(b-a) (x-b) (b-c) (b-d) . . (b - n) 
u c 

+ ; = 

(c-a) (c-b) (x-c) (c-d) (c-n) 

u d 

+ ; 

(d-a)(d-b)(d-c)x-d)(d-n)&etc. 



12. As an example, let it be required to find the logarithm of 
326, having given log 323=2.5092, log 325=2.5119, log 328=2.5159 
and log 329 « 2. 5172. We may write these values: 

Log 323 =u a =2.5092 and a=0 
Log 325 =u b =2.5119 and b=2 
Log 328 =u c =2.5159 and c=5 
Log 329 =u d =2.5172 and d=6 

x, the required term, is 3. We may now place these values in the 
formula and solve for the required value, u x . We have: 



LA GRANGE'S theorem. 17 



ux ux ux 



(3 -0) (3 -2) (3 -5) (3-6) (3) (1) ( -2) ( -3) 18 

ua ua — ua 

(3-0) (0-2) (0-5) (0-6) (3) (-2) (-5) (-6) 180 
ub ub ub 

= =— + 

(2 -0) (3 -2) (2 -5) (2-6) (2) (1) ( -3) ( -4) 24 
uc uc uc 

(5-0) (5-2) (3-5) (5-6) (5) (3) (-2) (-1) 30 
ud ud — ud 



(6-0) (6-2) (6-5) (3-6) (6) (4) (1) (-3) 72 

We may now write the derived values as an equation, thus: 

ux — ua ub uc — ud 

— = +— +— + 

18 180 24 30 72 

Reducing these fractions to their least common denominator, which 
is 360, and clearing of fractions, we have : 

20ux= — 2ua + 15ub+12uc(— 5ud) and solving 
-2ua =2. 5092X2=- 5.0184 
-5ud=2. 5172X5 = -12.5860 



-17.6044 

15ub =2.5119X15 =37.6785 

12uc = 2 . 5159 X 12 = 30 . 1908 



67.8693 

Uniting these sums algebraically, we have: 

20ux = 67 . 8693 - 17 . 6044 = 50 .2649. 
Dividing through by 20, we have: 
ux =2.5132, which is the required logarithm. 

13. In this formula, x is regarded as the unknown term, and in 
the example given is the 3rd term from 323 which is called ua. If 
the log of 327 had been required, x would be 4 and the solution would 
have had the same form, 4 taking the place of 3 throughout. In writing 
the denominator of ux in the formula each of the given values is de- 
ducted from x in succession. Thus (x— a), (x — b) and so on. In 
writing the denominators of the given factors ua, ub & etc., the same 
course is followed, that is, each given factor is deducted. Thus, to 
begin with a, we have (a — b), (a — c), (a — d) and (a — a). The latter 
is changed in producing the formula to (x — a) and is so written. Also, 



18 FINANCE AND LIFE INSURANCE. 

with b, we have (b— a), (b— b), written (x— b), (b— e) and (b — d) as 
the factors of the denominators of ub, and so with c and d. The 
peculiar value of this formula lies in the fact that equidistant terms 
are not essential in order to find the unknown term. 

14. The method of finite differences may be used also to find the 
sum of a series where the series follows some law of development 
and the differences vanish after a few orders or when the series is con- 
vergent. In the first case, the sum is an exactly, and in the last, gn 
approximately, correct summation, sufficiently exact for some pur- 
poses. The formula for this method is, 

n(n-l) n(n-l) (n-2) 

Sum or S=na-] AH A 2 + etc. 

2 |3 

15. Let it be required to find the sum of the series: 
1+54-15+35+70 + 126+ to 30 terms by the method of differences. 
Arrange the terms and differences as follows : 

Series A 1 A 2 A 3 A 4 Inserting these differences in the 

1+ 4+ 6+ 4+ 1 formula, we have: 

5 + 10+10+ 5+ 1 30-29 4 30 29 28 6 

15+20+15+ 6 30.1+ + : — + 

35+35+21 2 123 

70+56 30 29-28 27-4 30 29-28-27-26 

126 — + = 

1-2-3 4 12 3 4 5 

278256, Answer. 

In this example, the 4th differences are constant and the summa 
tion exact. 

16. Lubbock's Formula, for the approximate summation of a 
series of equidistant terms, may be employed when a single annuity 
value, ax, or insurance premium, Ax, is required. 

For Insurances, this formula may be conveniently written. 

Ax=n(uo+Ui+u 2 +etc.) — 

n-1 n 2 -l n 2 -l (n 2 -l)(19n 2 -l) (n 2 -l)(9n 2 -l) 

u + A A 2 + A 3 

2 12n 24n 720n 3 480n 3 

in which Ax is the present value of the insurance. 

For Annuities, it may be written 

n + 1 n 2 -l n 2 -l 

ax =n(u +Ui +u 2 +etc) 1 A h terms as above. 

2 12n 24n 



LUBBOCK'S FORMULA. 19 

17. The values u , Ui and etc., in the ease of insurances are repre- 

v n+1 dx+n 

sented by the general formula, ux+n = For annuities, 

lx 

v n lx+n. The 

they are represented by the general formula un= 

lx 
formula is an approximation in any event and third or fourth differences 
are sufficient for practical purposes. The co-efficients of the first, 
second, third and fourth differences are ( — .4), (.2), ( — .1264) and 
.0896, respectively, when n=5, and are ( — .825), .4125, (—2611) and 
. 1854 respectively when n = 10. Usually the result obtained by com- 
puting the value of u at intervals of ten years will be close enough. 

18. Let it be required to compute the approximate value, single 
premium, for an insurance of $1000 at age 35 on the American Exper- 
ience table and 4V2% interest. Here there may be six periods of ten 
years each from age 35 to the end of the table. First compute the 
values of an insurance of 1 payable at death during one year at each 
of these periods. First, we have d 35 v =732 Xv = . 956938^ (1 35 =81822) 

= .008560. Second, we have (d 43 =828) X(v u = .616199)-f- (I35 = 
81822) = . 006326, and so on to six terms. These results may then be 
arranged for differencing and summation as follows: 

A A 2 A 3 A 4 

u = . 008561 -002325 +001903 -001112 -001252 
ui = .006236-000422+000791 -002364 
u» = .005814+000369 -001573 
u 3 = . 006183 -001204 -n-1 

X .008561 = -.038525 

2 
u 4 = . 004979 n 2 - 1 

X - . 002325 = - . 001918 

12n 
u 5 = .001673 -n 2 -l 

X .001903= -.000785 

24n 
u 6 = . 000003 (n 2 - 1) (19n 2 - 1) 

X - .001112 = - .000290 

720n3 

.033449 -(n 2 -l) (9n 2 -l) . +000232 

X - .001252 =- 

10 480n 3 -.041286 



334490 
041285 



293204 X 1000 = $293 . 20, Answer. 



20 FINANCE AND LIFE INSURANCE. 

19. A number of formulas for the computation of special values 
of insurances or annuities are available, but they are not as a rule 
adapted to computations on the tables legally adopted in this country 
and the ones given here may be employed by the use of a little care and 
are deemed sufficient for this work. More occasions will be found 
for the use of Lubbock's formula by those for whom this book is de- 
signed, in the computation of annuities in connection with joint lives 
and this application of the formula will be further elucidated in treating 
that subject. Those unfamiliar with insurance notation might omit 
Articles 17 and 18 until after having read the chapters on annuities 
and insurance. 



CHAPTER IV 
Probability 

The doctrines of probability have frequent application in the solu- 
tion of problems depending upon life contingencies, and this chapter 
will be devoted to explaining and illustrating some of the principles of 
probability. 

1. The Mathematical Probability of the happening of an event 
has been denned as the number of favorable opportunities divided by 
the whole number of opportunities, favorable and unfavorable. 

2. Suppose, in common language, the odds are 3 to 2 against the 
happening of an event. Here the event may occur in two ways, the 
sum of the ways being 5. The probability of the happening of the 
event is two of these, or, 2 / 5 , and the chance of failure is 3 / 5 . The sum 
of these probabilities is 2 / 5 + 3 A =1, unity or certainty, since the event 
must either happen or fail. 

3. The principle may be extended to include any number of ways in 
which a single event may happen when one and only one can happen. 
Thus, suppose there are three applicants for a position and that A may 
obtain it in 2 ways, B in 3 ways and C in 4 ways, all of which are equally 
likely to occur. Here, there are 9 ways in which the event may be 
brought. to pass. A's chance is therefore 2 / 9 and his probability of 
failure '/•• That is, the odds are 7 to 2 against him. B's chance is 
3 / 9 and C's 4 /s>- The sum of the chances is 1. 

4. A familiar example illustrative of this subject is to suppose, say, 
3 white balls, 4 black balls and 5 red balls to be mixed in a bag from 
which balls are to be drawn at random and the chances determined. 
There being twelve balls in all and three possibilities that the first ball 
drawn may be white, the probability of drawing a white ball is 3 /i 2 . 
After the first ball is drawn, if it be a white one, the probability of again 
drawing a white ball is 2 /u. In the same way, it appears that the 
probability that the first ball drawn will be black is */i 2 , and that it will 
be red, 5 / 12 . 

5. Suppose it is desired to draw out the balls in pairs ; what would 
be the probabilities in the several cases? 

On the principles of permutations and combinations, it may be 
shown that the number of combinations of m things taken n at a 

time is . Therefore, the number of pairs which may be 

[ m-n | n 

12 11 

formed from 12 things taken 2 at a time is =66. The number of 

1-2 



22 FINANCE AND LIFE INSURANCE. 

3 2 

pairs that can be formed of the three white balls taken 2 and 2 is = 3 ; 

12 

43 5 4 

of four black balls =6, and of five red balls, = 10. The 

12 12 

probability of drawing two white balls is therefore 3 / 66 , which is also 
the product of the probabilities of drawing two white balls in succession 
which we have seen is 3 /i 2 and 2 /n. The probability of drawing two 
red balls is 10 / 6 6, which is likewise the product of drawing them in 
succession. That is 5 /i2 ■ 4 /n = 10 /66. 

6. Again, as the three white balls may each be paired with a black 
or a red ball, there may be formed 12 pairs consisting of white and 
black balls, 15 pairs consisting of white and red balls and 20 pairs 
consisting of black and red balls. From these may be derived the 
probabilities of drawing pairs, consisting of balls of the different 
colors. Thus, the probability of drawing a white and black pair is 
12 /e6, a white and red pair, 15 /ee, and a black and red pair, 2 % 6 . 

7. If we know the respective probabilities of the happening of 
two or more independent events, we may find the probability that all 
will happen by multiplying together the several probabilities. 

8. When the events are dependent, the probability that all will 
happen is the product of the probability of the first event and the 
probability that when the first has happened, the second will follow. 

9. Thus, in the example, the probability in the first instance of 
drawing a white ball was 3 /i 2 . Supposing the ball to be replaced, the 
probability of drawing a white ball would again be 3 /i 2 . The probability 
that a white ball would be drawn in both instances is ( 3 /i 2 ) 2 , or Vie. 
But we have seen that the probability of drawing the first is 3 / i2 and the 
second, without replacing the first, 2 /n. The probability of drawing a 
white ball in both cases is 3 /i 2 • 2 /n, or 3 / 66 . In the first instance, we deal 
with independent, and in the second, dependent, probabilities. 

10. The probability of throwing an ace with a single dice is 1 / 6 ; 
the probability of failing, 1 — 1 / e = 5 / 6 . The probability of throwing 
an ace twice in two throws is ( 1 k) 2 = l /m. The probability of failing 
twice in two throws is ( 5 /e) 2 = 2B /36, and the probability of not failing 
twice, that is, of succeeding at least once is 1 — 25 / 36 = ll /«. The prob- 
ability of throwing exactly n aces in a given number of throws will 
be noticed in the next article. 

11. Let p be the probability of succeeding, q the probability of 
failing. Now let it be required to find the probability of throwing an 
ace exactly three times in, say, 4 throws. By the principles of Articles 
4, 5 and 10, we may derive the following formula: 



MORTALITY AND PROBABILITY. 23 

n(n-l) (n-r + 1) 
P= pr q n ~ r which is general. In the present 

l£_ 
case, n=4 and r=3, so that the formula becomes numerically 
4 



4 3 2/1 V/5 \ 5 
l-2-3\6/\6/ 324 



12. Suppose it is desired to find in what number of trials the 
chance of throwing an ace becomes 1 / 2 , that is, an even wager. Let 
x = the number of trials. The probability of failing on each trial is 
6 / 6 and that of failing x times, that is, every time ( 5 /e) x and the proba- 
bility of succeeding is 1 — ( B /e)* This, by the conditions of the problem, 
must equal l A>, that is, 1 — ( 6 /e) x = V2. Transposing and uniting terms, 
we have ( 6 /6)* = 1 /2. Also x log 5 / 6 = log l /a- 

log 1 /* log 1 -log 2 .3010 .3010 

Andx= = = = =3.85, 

log 5 /e log 5 -log 6 .6999 -.7781 .0782 

which means that the chances of winning are a little better than one 
half in four throws. 

13. By a certain mortality table, 89685 persons are living at age 
30; at age 40, 82277 of them survive. The probability of one of the 
original number living ten years, supposing their chances of living to 
be equal, is the quotient found by dividing the latter number by the 
first, thus, 82277-r- . 89685 = .9397. At age 40, the probability of 
living ten years would be 72795^82277 = .8848. 

14. The probability of dying in ten years is the quotient found by 
dividing the number dying during the term by the number living at 
the beginning, thus, 7408-5-89685 = .0837. 

15. The probability of x, aged 30, and y, aged 40, living ten years 
is the product of the probabilities that each will live ten years, thus, 
. 9397 X. 8848 = .83144. 

16. The probability that y will live and x die in ten years is like- 
wise the product of the separate probabilities, thus, . 0837 X. 8448 = 
.074057. 

17. The probability that x will live ten years and die within the 
next ten years is the product of the probability of living the first ten 
years into the probability of his dying within the second ten years, 
thus, . 9397 X. 11512 =10816. 

18. The following notation will be extensively used in the succeed- 
ing pages and should be remembered, (x) denotes a person aged x 
years, (y) a person aged y years ; (1) denotes the number living according 
to a given mortality table, lx is therefore the number of persons living 
of a given age. So of ly, lz, & etc.(p) denotes the probability of living 
and unless modified by the use of other symbols, means the probability 



24 FINANCE AND LIFE INSURANCE, 

of living one year, (q) denotes the probability of dying and likewise 
involves only one year unless modified by other symbols. If n years 
are involved, the symbols are written npx or nqx, respectively, (p) 
also is to denote the probability of the occurrence of any event and (q) 
the probability of its not occurring. 

19. On principles already noticed, 1— p=q, and 1 — q=p, and 
p + q=l. The probability of two independent events happening is 
P1XP2 or pip 2 . The probability of failure is qiXq2 or qiq 2 . The 
probability of one person living and another dying is p • q. If the term 
be made n years, the probabilities are npxXnqy. (Art. 16.) The 
same principles apply to larger numbers of persons. 

20. The joint probability of living is pxy, meaning one year. If 
the term be n years, the form for the probability is npxy, which is equal 
to npx Xnpy or in terms of a mortality table, lx-fn ly+n. 

X 

lx ly 

21. The probability that both will die in n years is [nqxy (nqxX 
nqy) or in terms of the mortality table, lx — lx +n ly — ly +n 

X 

lx ly 

22. The probability that one, but not both, exactly one, of two 
persons will die in n years, is the sum of the probability that x will die 
and y will live and the probability that y will die and x live. That is, 
(nqx Xnpy) +(nqyX npx). In terms of the mortality table, this is 

(Ix-lx+n ly+n \ / ly-ly+n lx+n \ 
lx ly / \ ly lx / 

23. The probability that at least one of two persons will survive 
n years is the probability that both will not die in that time. That is, 
1 — | n qxy = npx + npy — npxy. 

24. The probability that x will die in any certain year is the 
product of the probability that he will survive the intervening term and 
the probability that he will die in the succeeding year. Thus the 
probability that x, aged 30, will die in his 40th year is in terms of the 
mortality table, n being 10. 

139 I39 — Uo 

X — — — 

I30 I30 

25. The probability that one of two lives will fail in the nth year 
is the probability that both will survive n — 1 years, less the probability 
that both will survive n years. In terms of the mortality table 

I39XI49 U0XI50 



I30XI40 I30XU0 
The foregoing is sufficient to illustrate most of the processes employed 
in this book in the solution of problems involving the element of chance 
or probability. 



CHAPTER V 
Compound Interest 

1. Wlien interest is added to the principal and interest computed 
upon the amount as a new principal, the interest is called Compound 
Interest. Most of the operations in finance involve compound in- 
terest. 

2. Tables of Compound Interest computed on a principal of one 
for a term of sixty periods at rates ranging from 5 /s of one per cent to 
ten per cent and for one hundred and twenty periods at rates ranging 
from one-fourth of one per cent to one-half of one per cent are published 
in this book, the results given being the amounts of one improved at 
the given rates. 

3. The Compound Interest upon a given principal for a given rate 
is found by multiplying the given principal by one plus the rate for 
one period and this product by the same multiplier, and so continue for 
each period in the term. From the last product, deduct the principal. 
The remainder will be the compound interest. 

4. If we represent the rate by the letter i, the principal by P, the 
term or number of periods by n, the amount of principal and interest 
by S and the compound interest by I, we may from Section 3, develop 
some useful formulae. 

5. Thus, we notice that the amount for the first period is 1-fi; 
that for the second period is (1 +i) (1 -H) = (1 +i) 2 ; that for the third is 
(1 -fi) 2 (l +i) or (1 -f-i) 3 and generally the law of the accumulation being 
apparent, S = (l+i) n . (1) 

6. From Article 5, it will be apparent that logarithms afford a 
very rapid and accurate means of producing a table of amounts at 
compound interest. Thus, in computing, for instance, the compound 
amounts of 1 at one-fourth of one per cent, it was only necessary to 
put on the adding machine the log of 1 .0025 to eight places of decimals 
and add continuously to 120 places, recording the amount at each pull 
of the handle. The anti logs to six places is the table desired. 

7. Transformations of the equation S = (l+i) n will enable us to 
find the other factors involved in the computation, thus : 

S = (1 +i) n , Log S =n log (1 -f-i) and 

logS 
n= (2) 

log(l+i) _ 

Again, S = (l+i) n , D VS = l+i and 

LogS 
i= n VS-l= 1 (3) 



26 FINANCE AND LIFE INSURANCE. 

8. If instead of 1, we introduce the principal P, formulas 1, 2 and 
3 become respectively, S=P(l-fi) n (1) 

log S-log P 

n (2) 

log (1+i) 

/S Log S- Log P 

i== u^ 1== J (3) 

P n 

9. From the equation S=P(l+i) n , the value of P is derived as 
follows : 

1 

S=P(l+i) n andP=S-Hl+i) n =S also P = Sv* (4) 

(l+i) n 

10. A numerical example of the applications of each of the four 
equations, just derived will sufficiently illustrate their meaning. Let 
P=500, n=8, i = .05. First, to find S, we have 1+i =1.05 
lX(l+i)=1.05 
(1) multiplied by 1.05 = .1525 



(2) multiplied by 1.05 = 

(3) multiplied by 1.05 = 

(4) multiplied by 1.05 = 

(5) multiplied by 1.05 = 

(6) multiplied by 1.05 - 

(7) multiplied by 1 . 05 = 



(8) multiplied by 500 =738.728 = P(l+i) 8 

Log S — Log P 

Again from Art. 8, n = and from the table of logs 

Log (1+i) 
we have Log 738 . 73 = 2 . 868486, Log 500 = 2 . 698970, Log 1 . 05 = .021 189 
2.868486-2.698970 

and = 8. Hence n = 8 years or periods. 

.021189 
log S -log P 
Also i = 1 and from the figures just found we have log 



1.05 
.1525 


= 


(1+i) 

d+i) 2 

d+i) 8 

(1+i) 4 

(1+i) 5 

(1+i) 6 

d+i) 7 

(1+i) 8 


(1) 


1 . 1025 
55125 


(2) 


1 . 157625 

57881 


(3) 


1.215506 
60775 

1.276281 
63814 


(4) 
(5) 


1.340096 
67005 


(6) 


1.407101 
70355 


(7) 


1.477456 


(8) 



INTEREST AND DISCOUNT. 27 

P- log S = . 169516, which divided by 8, gives .021189, the log of 
1 . 05. Subtracting 1 , we have i = . 05, which we know to be correct. 

1 

Finally, p=S or S v n . From Art. 10, we already have 

d+i) n 
(1 +i) n =1 .477456. Dividing 1 by the letter sum using the contracted 
method, thus : 1 . 477456) 1 . 0000000 ( . 676839 

8864736 



1135264 
1034219 

101045 

88647 

12398 
11819 



579 
443 

136 
132 

This quotient is the function v n and is the present value of 1 due in 8 
years, money being worth five per cent. 

Multiplying S or 738.728 by .676839, using the contracted method, 
we have P, thus : 738 . 728 

938.676 



4432 368 


517 110 


44 324 


5 910 


222 


66 



500.0000 

11. While the unit of time in quoting rates of interest is usually 
considered to be one year, it is convenient to regard the tabulated 
values as referring simply to periods. By this means, the tables may 
be used to compute the amounts where the interest is payable semi- 
annually, quarterly or at other fractional periods of a year. Thus, 
had the problem of Article 10 prescribed interest payable semi-annually 
a term of 8 years would have comprised 16 periods and the table of 
compound amounts at 2V2 per cent should be used in the computation. 
Or if direct computation be resorted to, the formula used would be 

S=P(l+i/ 2 )*n (5) 

12. It is proper to explain that the results reached in the two 
cases are not identical for the reason that by formula (5), one-half of 
the annual interest is paid each year six months before the end of the 



FINANCE AND LIFE INSURANCE. 



year and is compounded at one-half the annual rate so that the lender 
actually receives 5 . 063 per cent interest on his principal. It is believed 
to be the general practice, however, to compute the interest as stated 
in Article 11 and if the transaction is understood by the parties, no 
injustice is done and the great inconvenience which would be occasioned 
by having to ascertain and base all calculations upon the effective rates 
is thereby avoided. 

13. The following short tables of nominal and effective rates are 
taken from Mr. George King's book, Theory of Finance. In the 
tables this effective rate is designated as i and the nominal rate as j, 
in accordance with the notation used by Mr. King. 



1 


2 


3 


4 


5 


6 


Nominal 


Effective 


Effective 


Effective 


Nominal 


Nominal 


Rate j 


Rate i 


Rate 


Rate i 


Rate 


Rate 




Semi- 


Quarterly 




Semi- 


Quarterly 




Annual 






Annual 




.025 


.025156 


. 125236 


.025 


.024846 


.024369 


.03 


.030225 


.030339 


.03 


.029778 


.029668 


.035 


.035306 


.035462 


.035 


.034699 


.034550 


.04 


.040400 


.040604 


.04 


.039608 


.039414 


.045 


.045506 


.045765 


.045 


.044505 


.044260 


.05 


.050625 


.050945 


.05 


.049390 


.049089 


.055 


.055756 


.056145 


.055 


.054264 


053901 


.06 


.060900 


.061364 


.06 


.059126 


.0*58695 


.065 


.066056 


.066602 


.065 


.063976 


.063473 


.07 


.071225 


.071859 


.07 


.068816 


.068234 


.08 


.081600 


.082432 


.08 


.078461 


.077706 


.09 


.092025 


.093083 


.09 


.088061 


.087113 


.10 


. 102500 


.103813 


.10 


.097618 


.096455 



14. Illustrating the use that may be made of the tables of Article 
13, let us suppose that the rate quoted is 4 per cent. If interest is 
made payable semi-annually (by Column 2), the principal will earn 
4.04 per cent per annum, and, if payable quarterly (by Column 3) 
it will earn 4.0604 per cent per annum. Again, if it is desired to earn 
exactly four per cent per annum, the rate which must be quoted 
payable semi-annually is (by Column 5) 3 . 9608 per cent and the rate 
which must be quoted payable quarterly is (by Column 6) 3 . 9414 per 
cent. 

15. The effective rate may be derived from the nominal rate by 
formula (5), n being taken as one year and the principal as 1, thus 
i = (l+7p) p (5a) 

Where p is the number of times interest is convertable in one year. 

16. The nominal rate, or rate to be quoted, payable semi-annually, 
quarterly or at some other fractional part of a year may be computed 
by the following formula : 

j=p(l+i)7p-l or pVTH- 1 (6) 



VALUATION AND RATES. 29 

17. Thus, suppose it is desired to realize 5 per cent per annum 
upon a rate to be quoted payable semi-annually. Applying formula 
6, we have: 



2(V 1.05-l)=j. The square root of 1.05 is 1.024695. Sub- 
tracting 1 and multiplying by 2 we obtain .04939, which is the rate 
given in the table. If a rate payable quarterly is required, the 4th 
root of one plus the effective rate is taken, one subtracted and the 
remainder multiplied by four. The fourth root may be found by taking 
the square root of the square root, or logarithms may be employed. 

18. It is sometimes desirable to know what rate of interest per 
annum will be realized upon a bond or note bearing interest payable 
annually, semi-annually or quarterly, where it is bought at a discount 
or premium. In such case, the amount on the principal named in 
the instrument is found at the prescribed rate for the term. Then the 
price paid for the bond or note may be taken as the principal for the 
purpose of finding the rate, and formula (3) is used in the solution, 
thus: 

19. Let it be required to find what rate will be realized upon a 
bond of $1000.00, bearing six per cent interest payable semi-annually 
and running ten years, for which $1100.00 is paid. First, compute 
the amount on the principal, $1000.00, for 20 periods at 3 per cent, 
or from the table, No. 2, we find it to be $1806.00. We now assume 
the latter sum to be the amount of an investment of $1100.00 at the 
end of ten years, and for the purposes of the formula S = 1806. 11, 
P = 1 100 and n = 10. Log S = 3 . 2567442, Log P = 3 . 0413827 and Log 
S -Log P-f- 10 = .0215352 = 1 .05087. Deducting 1, we have the rate 
5.09 per cent. 

20. Conversely, it is often desirable to know what sum may be 
paid for a bond or note to realize a given rate per cent on the invest- 
ment. In this case, the problem is to find the principal, the amount 
being known, or ascertainable, from the terms of the bond. Formulas 
1 and 4 may be employed, but the factor v n is computed at the rate 
intended to be realized, thus: 

21. Let it be required to find what price may be offered for the 
bond described in Article 19 so that seven per cent per annum may be 
realized on the investment. First, the amount of the bond at the end 
of the ten years we have found is $1806 . 11. v 10 computed at the rate 
of 7 per cent in the manner illustrated in Article 10, or also as given in 
the table, is .508349. In the formula P = Sv n , the first number 
1806 . 11 corresponds to S and the latter, . 508349 to v n and their product 
$918.13 is P. The latter sum therefore is the price which may be 
paid. 

22. Formula 2 provides a method by means of which we may find 
in what time a principal will double or otherwise multiply itself at a 
given rate per cent, thus to find in what time money will double itself 
in the formula let 2P = S; to quadruple itself, 4P = S, and so on. 



30 FINANCE AND LIFE INSURANCE. 

23. Let it be required to find in what time a debt of one dollar 
will amount to two dollars at three and one-half per cent compound 
interest. Let S=2, P = l and i = .035. Log S = . 30103, Log P=0, 
Log 1 . 035 = . 014940. Then 

Log S— Log P 

=20 . 1 years. 

Log (1 +i) 

The quotient is taken as a natural number, not a logarithm. 

69 

24. It may be easily shown that (- . 35 is a reliable approximate 

i 
formula for finding the time in which money doubles itself at compound 
interest. That is, divide 69 by the given rate and add . 35 to the quo- 
tient. 

25. If S be taken as 1, then P, according to formula 4, will be 

1 1 

v n , or if the term be one year, the first number becomes 



(l+i) n 1+i 

which is called simply v. This is the present value of 1 for one year. 
v n is the present value of 1 for n years. The process of finding the 
present value of 1 is called discounting. This present value cor- 
responds to P representing the principal in the operations of compound 
interest and its value is the reciprocal of S, that is, of the amount of 
the principal and interest. 

26. A few numerical examples of the principles of Article 25 will 
help to elucidate the subject. If we assume S = l, i=5 and n = l, 
then v = 1-M. 05 = .952381. If interest be computed on .952381 at 
5%, the amount will be 1. Hence, .952381 is the present value of 1 
at 5%. If the present value be deducted from 1, we have .047619. 
This is called discount and is usually designated by the symbol d. 
If we change n in the first example to say, 20, then by formula 4, we 

1 

have the equation P = 1-r- (1 . 05) 20 . The latter quantity will be 

(1+i) 20 

recognized as the amount of 1 in 20 years at 5 per cent, and by the table 
is 2 . 653298. Performing the indicated division, we have v 20 = . 376889. 
Subtracting the latter sum from 1, we have .623111 which may be 
appropriately called the compound discount on 1 in 20 years at 5 
per cent. From the foregoing, the remark is justified that v and P 
are equivalents in that they are the present values of an amount. 
The following additional formulas may be drawn from the foregoing 
discussion: 



1 




v= 




1+i 




d = l-v 




1 




v«= 


-or (l+i)- n 



DISCOUNTING. 31 

(7) 
(8) 
(9) 



(l+i) n 



1 


1 


1 


d+i) 1 
1 


A 

1 d+i) 
-X(l+i) = 


(l+i) n+1 
1 



27. It has been noticed that the denominators in the right numbers 
of equations 7 and 9 are compound interest amounts. They are 
therefore powers of 1 +i and since they increase as the ratio, 1 +i, 
is raised to higher powers, the numerator remaining the same, the 
function v n decreases, thus: 



, or, v n Xv=v n+1 . On the other hand, 



or v n_1 . 
(1+i)- (1+i) 11 - 1 

We may therefore make a table of v n by either of the two methods 
but the latter would be simpler, since 1 +i is an easier multiplier than 
v. As in the case of compound interest, logarithms afford the best 
means of producing a table of v n and such was the method used in 
computing the tables of this book. First, find (1 +i) 120 at X U%. Take 
its log and deduct from the log of 1, which is or 10 — 10, using eight 
places of decimals. Put this remainder on the adding machine and 
add successively the log of 1 . 0025, recording the sum at each pull of 
the handle. The anti logs to six places will be the result sought. 

28. For short terms, it is practicable to make the table by actual 
multiplication. Thus for v 20 at say, 5 %, we may begin with v = . 952381 
and multiply successively by .952381 to 20 periods. The contracted 
method will be best. We may compute the amount of 1 at compound 
interest at 5 per cent, to twenty periods, divide it into 1 and use 1 .05 
as a multiplier. We have 1-f- 2 . 653298 = . 376889 = v 20 . Then, 

.376889 = v 20 
18845 



v 20 Xl.05 = .395734 = v 19 
19787 



\-i 9 Xl.05 = .415521 
20776 



v^xl.05 = .436297 = v 1 



32 FINANCE AND LIFE INSURANCE. 

and so on, it being only necessary to use 5 in the actual multiplication 
being careful to carry into the right figure of the product the proper 
number from the two rejected figures. Of course, if a compound 
interest table is available, the v n table can be made directly from that 
by formula (9). 

29. Conversely, a compound interest table may be derived directly 

from a table of v n . Thus, from the equation, 1 

v n = we have 

d+i) n •• 

v n (l+i) n = l and finally, 

1 (10) 

(l+i)n=_ 



By formula 10, therefore, we may find at once the compound 
amount on any number by simply taking the reciprocal of the present 
value of the same number for the same time and rate of interest. 

30. A compound interest table may be used to find the compound 
interest for a term extending beyond the term of the table on the fol- 
lowing principle: The amount of 1 for n periods, by equation (1), is 
(1 -fi) n - Likewise, the amount of 1 for m periods is (1 +i) m , the expo- 
nents being considered general. And by the theory of exponents, 
(l+i) m+n = (l+i) m X(l+i) n . Therefore, if it be desired to find the 
compound interest on 1 at say 3% for, say 110 years, we have simply 
to multiply the amounts for fifty and sixty years together, thus: 

4.383906 

3 061985 



21 919530 


3 507125 


394551 


4384 


2630 


13 



25.28233 

In the multiplication, for convenience, the figures of the multiplier 
are reversed. To find the interest, deduct 1 from the product. If the 
principal be other than one, multiply the result for 1 by the principal. 

31. The table of v n is governed by the same principles and may be 
extended in exactly the same way. An example is not necessary. 

32. We have seen, Article 11, that S=P (l+i/m) mn represents 
the compound amount of 1 for n years payable fractionally. If the 
term be made one year and the principal 1, then the interest for one 



INTEREST AND DISCOUNT EXAMPLES. 33 

year would be represented by the expression (l+i/m) m — 1. The 
quantity, m, may be supposed to increase without limit, in which 
case, the interest would be treated as accruing continuously or mo- 
mently. Now, if we suppose m, in the expression (1 +i/m) m to increase 
without limit, the expression by the theory of logarithms has e 1 for its 
limit, e being the base of the Napierian system of logarithms and i the 
nominal rate of interest. The effective rate then becomes e' — 1 from 
which we have the formula i =e* — 1. (11) 

Thus, to find the effective rate of interest convertible momently 
the nominal rate being, say 4 x / 2 %, we have e 1 =4 1 / 2 log. 2.71828 = 
(. 045 X. 43294) = .0195432. The anti log of the last quantity is 
1.0460277, from which we deduct 1 according to formula 11, leaving 
as the effective rate 4.6028 per cent. 

33. Again, by simple transformations of formula 10 we derive a 
formula for the nominal rate, sometimes called the force of interest 
represented by the symbol B or d', thus: 

i =e'— 1. e'=i +1 or i into log e =log i +1 and finally, 

log (i +1) log (1+T) 

d'or S= = (12) 

log e .43294 

It may be proper to remind the reader that as used in the last two 
articles, 2.71828 is "e" of the Napierian System of logs and .43294 is 
the modulus of the common system. 

34. In the same way the value of v and d' may be found when 
the discounting is regarded as progressing continuously or momently 
and we have: v =e- d ' (13) 

35. An approximate rule for finding the force of discount or 
force of interest is to take one-half the sum of the effective rates of 
interest and discount, the result will be the continuous or momently 

rate. Thus, i+d (14) 

d'= 



This formula is accurate enough at all usual rates and dispenses with 
the use of logarithms where interest and discount tables are available. 
For instance, to find d' at the rate, 4 1 / 2 %, we have from the v n table 
v = . 956938 and d = 1 - .956938 = .043C62 and (.045 + .C43C62)H-2 = 
.04403. 



CHAPTER VI 
Of Annuities Certain 

1. An annuity certain is a fixed sum payable at the end of each 
period of a fixed term. It is distinguished from a life annuity by the 
fact that the term of the latter is fixed or affected by the duration of a 
given life or certain lives. Ordinarily, the payments of an annuity 
are due at the end of the periods, usually one year, but they may be 
contracted to fall due at the beginning of the year or period. Such 
an annuity is called an annuity due, and its symbol is the ordinary 
letter a. 

2. If 1 be made payable at the end of one year its value now, as 
we saw in last chapter, is v. If it be payable in two years, its present 
value is v 2 ; in three years, v 3 , and so on. That is, the values are 

1 1 1 

, , and so on. 

1+i (1+i) 2 (1+i) 3 

From this, we see that the present value of 1 a year for 2, 3 or n, that is, 
any number of years, is v+v 2 +v 3 + * * +v n . Such a sum is the 
present value of an annuity of one for n years, and is often represented 
by the symbol an|, the character n| indicating any fixed or certain 
term of years or periods, and the letter a being printed in italics. 

3. On the principles of Article 2, a table of the present values of 1 
payable in a series of years or periods may be made by summing the 
values of v, v 2 , v 3 and so forth, beginning with the value v and recording 
the sums at each step. Thus to produce a table on| at six per cent, we 
may first write a line of v n and then sum across the page thus: 





v n 


«n| 


(1) 


.943396 


.943396 


(2) 


.889996 


1.833392 


(3) 


.839619 


2.673011 


(4) 


. 792094 


3.465105 



adding (1) of the second line to (2) of the first and so on. 

4. The value of any particular annuity may be found directly 
by the formula 1— v n 

a n| = (15) 

i 
Referring to Article 2, it will be noticed that a n| is the sum of a 

1 1 

geometrical series of which — is the first term, the second and 

1 + i (1+i) 2 

1 1 

the last. Also that the ratio is . 

(l+i) n 1+i 



ANNUITIES— FRACTIONAL PAYMENTS. 35 

1 



1 

(l +i )n i-.vB, 

The value — or its equivalent, will at once be recognized 

i i 

as the sum of jthis geometrical series. Its reasonableness will also be 
appreciated if we consider that if 1 be invested for a term of n years 
it will produce i interest each year and the original capital, 1, will 
remain to be returned to the investor, the two amounting to (l+i) n . 
Now it is apparent that 1 is the present value of the amount (l-fi) n 
because it is the value that produced it. Also, we know that v n is 
the present value of 1 in n years and if we deduct v n from 1, the remain- 
der will be the present value of an annual payment of i for n years. 

Then, by proportion, i:l as 1 — v n : 1 — v n 



5. We saw in the last chapter, Articles 25 and 26, that v n is the 
reciprocal of (1 +i) n . This fact gives us a hint as to the proper method 
of dealing with annuities having fractional payments and based on 
interest convertible at fractional periods of a year. Thus, if 

(p) 
(l+i)" n =v n , (l+i/p)- n p, also equals v n P or v n as it is more correctly 

written. 

6. We may now write down formulae expressing in a concise way 
the value of annuities payable fractionally and based on interest com- 
pounded at fractional periods of the year. Let m be the payment 
periods, p the number of times money is convertible each year and n 
the term of the annuity in years. We then have the general expression: 

1 — (1 +i/p)- r 'p 

a n| = . (16) 

m(l+|)P/m 

7. The operations indicated in the right member of equation 16 
ma}* be stated in the form of a rule as follows: First, find the value of 
the numerator by raising 1 plus the fraction of the rate prescribed to 
a power equal to the number of periods in the whole term. Divide 
1 by this amount and subtract the quotient from 1. Second, find the 
effective annual rate for one year at the given rate of interest convertible 
as prescribed in the problem ; then from this find the nominal rate for 
one year with payments as prescribed in the problem. Third, divide 
the value first found by this rate. A few examples will illustrate the 
application of the formula and rule and also indicate the effect of 
differences in the interest periods and payment periods. 

8. Let it be required to find the value of an annuity of 1 for ten 



36 FINANCE AND LIFE INSURANCE. 

years at six per cent interest convertible quarterly with payments 
one-half each half year. Applying the formula, we have : 

d+V4)- 4n 
1 



2(.(I+i/4)Vi-l) 

From the table, we obtain the value of (1 .015) 40 = 1 .8140. Dividing 
1 by this value, we have .551267. Subtracting the latter amount 
from 1, we have for the numerator .448732. Then ( (1.015) 2 -1) = 
.030225. Multiplying by 2, we have for the rate .06045. Finally 
dividing .551267 by .06045, we have the value of the annuity, 7.4232. 

9. Let it be required to find the present value of an annuity for 
25 years at 4 per cent convertible four times a year, with payments 
1 / 4 each quarter. First, computing the compound interest for 100 
periods at one per cent, we have 2 . 704814. Taking its reciprocal as 
before, we obtain . 36971 L Subtracting the latter sum from 1, we have 
. 630289. The fourth root of the fourth power of 1 . 04, we know, is 
1 .04, and the rate of interest is therefore 4. Dividing . 630289 by .04, 
we have the answer 15 . 7572. 

10. Let the conditions of the problem of Article 8 be so changed 
that the interest shall be compounded semi-annually, and the payments 

1 

be due l / 4 each quarter. . 553676. And 1 - . 553676 = . 446324. 

(1.03) 20 

This is the numerator. Also 4 ( (1 +3) 2 A-1) =4(V 1.03-1) =.059556 
This is the denominator of the formula. Performing the division as 
indicated in the formula, we have the answer 7 . 4942. 

11. The value found by formula 16 is that of an annuity of 1, the 
value of any other annuity may of course be found by multiplying the 
value of a n| by the given annuity. 

12. The value of an annuity due may be found by computing the 
value of an annuity for a term one less than the given term and adding 
1 to the result. Or take from the table an annuity for a term 1 less 
than the given term and add 1. Thus an annuity due of 1 for 20 years 
at 4% is 13.^33939 + 1. 14. 133939 = aT9| +1. From this we get the 
equation, a n| = a n — 1| +1. (17) 

13. An annuity to begin at some future date is called a deferred 
annuity and is of ten represented by the symbol m|an| =v m on|. (18) 

' 14. Let it be required to find the value of a ten-year annuity to 
be entered upon five years hence, at 5 %. From the table, we find the 
value of a ten-year annuity at 5%, to be 7.721735. But this value is 
not due for five years and must be discounted five years at 5%. 
That is v 5 , that is by .783526. Then .783526X7.721735=6.050180. 

15. When an annuity table is available, the value of a deferred 
annuity may be found more directly by finding an annuity for the 



FORBORNE ANNUITIES. 37 

sum of the two terms and deducting therefrom the value of an annuity 
for the intervening term. Thus, to solve the problem of Article 14, 
from fll5| = 10 . 379658, deduct ao\ = 4 . 329477 = 6 . 05018 1. From this 
we have the equation m |an| = am-}-n| — am\ (19) 

1G. Let us suppose that instead of receiving the annual payments 
of 1, the annuitant allows them to accumulate for n years at the pres- 
cribed rate of interest. What is its value? First, if we have the present 
value of the annuity, an|, we may treat it as the principal of an invest- 
ment and solve by equation 1 of the preceding chapter, S=P(l+i) n , 
and substituting an] for P, we have the formula: 
SH|=on| (l+i) n (20) 

17. The amount (1 +i) n consists of the 1 together with an annuity 
of i, the annual interest, and its accumulations for n years. If the 1 
be deducted, the remainder, (l+i) n — 1 will be the amount of the 

annuity i. Then by proportion, i: 1 as (1 +i) n — 1: (l+i) n — v 

, from 

i 

which we derive the formula: 

SE! = (l+i) n -l 

(21) 



This may be stated in the form of a Rule : Divide the compound 
interest for the given term by the interest for one period of the term. 

18. In cases where the annuity is payable, a certain part at the 
end of each corresponding part of a year, it is necessary to modify the 
denominator in equation (21) to correspond with the nominal rate, 
so that the formula for the amount of an annuity payable 1/m each 
1 'mth part of a year becomes: 

(m) (1+ir-l 

Sn|= (22) 

ma+iWm-l) 

19. If the problem involves also interest convertible at fractional 
periods, 1/p th, of the year, a corresponding modification of the last 
equation must be made, thus, 

U+7p) p >-i 

Snl = (23) 

p(i+Vp) p /^-i 

20. A few examples illustrative of the last four equations will be 
helpful. First, let there be an annuity of 1 per year for ten years, 
what will be its amount at the end of the term at five per cent interest? 

From the table, we have a!0\ =7.721735 and (1 .05) 10 = 1 .628895 
and by formula 20, the product of these =S10|. Performing the 
multiplication, we have the answer, 12.577894. 



38 FINANCE AND LIFE INSURANCE. 

Again, the amount of (1.05) 10 is 1.628895. Subtracting 1 and 
dividing by . 05, as indicated by equation 21, we have S . To| = . 628895-r- 
.05 = 12.5779 as before. 

Again, suppose the payments are to be made x \% each half year, by 

formula 22, we have (1 . 05) 10 - 1 . 628895 

= = 12 . 73326. 



2(V 1.05-1) .04939 

Suppose, also, that to the problem as last stated, the condition is 
added that interest shall.be convertible quarterly. By equation 23, 

(1.0125) 40 -1 .643619 

= = 12 . 8181. 

2(1.0125) 4 / 2 -l .050212 

Finally, let the conditions be that the interest shall be convertible 
quarterly, the payments being due annually. By equation 23, we have 

(l+74) 40 -l (1.0125) 40 -! .643619 
= = ■ = 12.6336. 



(l+VO 4 / 1 -! (1.0125) 4 -1 .050945 

21. We have seen, Article 20, that Sn|=an| (l+i) n . By a 
simple transformation of this equation, we have the formula 

Sn| 
an| = — — (24) 

d+i) n 

22. Let us suppose Sn| = 1. Let it be required to find what annual 
payment will produce it. Obviously, it is that part of 1 obtained by 
dividing it by Sn|. Thus, if SH)| at 5% = 12.5779 being the accumu- 
lations of 1 a year for ten years, then 1-h 12.577894 = .079505 is the 
annuity which will amount to 1 in ten years at 5 per cent. This func- 
tion is called a Sinking Fund. 

1 

It may be computed by the formula, Sinking Fund = (25) 

Sn 

23. A table of sinking funds may be readily computed from a 
table of SnJ by means of logarithms. Take the logs of a column of 
Snj. Take the arithmetical complement or co log of the latter and 
the anti log will be 'the sinking fund. 

24. The sinking fund table may also be conveniently formed by 
dividing the interest for a single year or period by the compound 
interest for the given term and rate, thus 

i-r-((l-fi) n -l) = sinking fund. (26) 

25. We have seen, Article 21, that Sn| is composed of 1 plus the 
annuity i and its accumulations. Hence, if we add the annuity i to the 
annuity which will produce 1 in n years, that is, to the sinking fund, 



SINKING FDNDS. 39 

we will have an annuity the present value of which is 1. Thus, we 
obtain the annuity which 1 will purchase. 

26. We may also find the annuity which 1 will purchase by con- 

1 

verting 1 into an annuity. This is, by dividing it by an| = — 

on 

27. An annuity represents an investment repayable in equal 
annual installments for a given term. The function on| is the prin- 
cipal invested and the annuity contains not only the principal but also 
the interest on the same. That is, at the end of the term both the 
principal and the interest will have been repaid by the annual pay- 
ments. It is sometimes important to know how much of the install- 
ments of a loan repayable by equal installments is principal and how 
much is interest. A hint as to the method of doing this will be found 
in Article 25, where it is noticed that an annuity, is composed of two 
elements, consisting of a sinking fund which will produce its present 
value and an annuity of the interest on the present value. The 
present value of each of these as the payments progress will represent 
the amount of each remaining unpaid. Naturally as the debt is re- 
duced the portion required as interest decreases and the part going 
to discharge the principal increases. 

28. As an example, let it be required to separate the installments 
of a debt of $1000.00 into principal and interest contributions, sup- 
posing ten equal installments in ten years, interest at five per cent. 

First, dividing 1000 by al0| as found in the 5 per cent column of 
the table, that is, 1000 by 7.721735, we have $129,505 as the annual 
installment. From the sinking fund table we find .079505 per annum 
will produce 1 in ten years at 5%, hence, 79.505 is required to produce 
1000 in the same time. This strikes one as being proper because we 
know that 50 . 00 is the interest for the first year. This is also in ac- 
cordance with Article 25. Hence, the first payment is composed of 
50 . 00 interest and 79 . 505 for sinking fund. The sinking fund for the 
next year with 5 per cent on the last year deposit is 83 . 47. Hence, 
for the second year, we have $129.50 -$83 .47 = $46.03 for interest 
and $83.47 for sinking fund. We may see that the interest $46.03 
is correct, by computing the interest on $920.50, the principal less the 
sinking fund. Again (920.50-83.47) X .05 =41 .85 is the interest 
contribution for the third year and 129.57 —41 .85 =87.66, the sinking 
fund. And thus a schedule may be made of payments, interest and 
sinking fund, for an installment loan. 

29. The amount in a sinking fund may be found at any time by 
treating the annual contributions to the sinking fund as an annuity 
accumulated for a term equal to the number of payments made. Thus, 
at the end of the 6th year, the sinking fund amounts to S6| =6.8019 X 
79.51 =$540,785. The amount of the debt remaining unpaid at the 
same date is, of course, 1000-540.79 =$459.21. 



40 



FINANCE AND LIFE INSURANCE. 



30. If the conditions of the problem of Article 28 be so changed 
that an equal installment of the loan plus the interest shall be paid 
each year, the schedule may be made up as well by direct computation 
as otherwise. The annual installment of principal is of course, $100.00 
and the work may be arranged as follows: 



Years 


Principal 


Installment 


Interest 


Total 
Payment 


1 
2 
3 


1000 
900 
800 


100.00 
100.00 
100.00 


50.00 
45.00 
40.00 


150.00 
145.00 
140.00 



and so on. 

31. A case may arise where the sinking fund bears a less rate of 
interest than the bond calls for. This presents a different situation 
from that discussed in Articles 27-29. Thus we may suppose the 
sinking fund in the problem of Article 28 to be deposited in a bank 
or trust company at, say, 3\/2 per cent until the bond matures, the 
annual interest to be paid the bond holder. If we keep in mind the 
fact that the purpose of the sinking fund is to accumulate a sum 
sufficient to pay the principal of the debt at the end of the loan period, 
the solution is simple. In that case, we need only to take the annual 
payment from the 3 x /2% sinking fund column, and in the present 
case, it will be .087231 X1000 or 87.231 per annum. Adding $50.00, 
interest, the annual sum to be paid by the debtor is $137.23 instead 
of $129.51 as in the problem of Article 28. 

32. The annuity tables enable us to solve problems similar to the 
one given in Article 21 of last chapter, providing the interest used in 
valuation is convertible at the same periods as the rate named in the 
bond. Thus, let it be required to find the price at which a ten year 
bond for $1000 .00 bearing four per cent payable semi-annually may be 
sold to earn six per cent, payable semi-annually. The present value 
of $1000.00 payable in ten years at 6% semi-annually as given by the 
table, v 20 of the 3% column, is, $553,676. The present value of a 
semi-annual annuity of $20.00 for 20 periods, found in 3% on| table 
is 14 . 87748 for 1 , which multiplied by 20 = 297 . 50. The sum of these, 
553. 68 +297. 50 =$851. 18 which is the value of the bond or price 
that may be paid for it to earn 6% semi-annual interest. 

33. The rate at which an annuity has been accumulated 

may be found directly from a table of annuities by running along 
the line opposite the given term until its value is found, and taking 
the rate given at the top. Thus, we may find the rate of a 20-year 
annuity, the present value of which is 13.00794 by following the 
horizontal line opposite 20 until we strike the value under the rate 
4V:%. 



FORMULAS FOR RATES. 41 

34. In eases where the annuity has not been computed upon any 
of the usually tabulated rates, a more difficult problem is presented. 
Two formulas will be given, one such as is usually found in books on 
interest and one from Mr. King's book, attributed to Mr. G. F. Hardy. 

35. If we let an| be the value of the given annuity, an| the value 
of an annuity for the same term taken from the same table and beiDg 
as near as possible to the value of the given annuity, j the rate of a' n| 
and p +j the required rate. The problem then is to find the value of 
p. For this the formula has been given: 

j(a'-a) 
P= (27) 



36. Let us take an annuity of 1 for twenty years, the present 
value of which is 12. 1261. What is the rate per cent? Looking into 
the table of an| opposite 20, we find 12 . 4622 as produced by the rate of 
5%. From the table v n we get v 21 at 5%, .358942. Substituting 
these values in equation, (27) we have the following: 

.05(12 4622-12.1261) 

P= 

12. 4622- 20 X. 358942 

Solving the equation, we have . 016805-v- 5 . 2834 = p = . 003181. Finally 
.05+003181 = .053181 which is the approximate rate required. 

37. The formula of Hardy is as follows : 

h(A-V*A s V2A 2 )- 1 

p = + (28) 

a-ai A-V2A 2 

The formula involves the processes of differencing and will be best 
understood by following its application in the solution of a problem. 
Let the example of Article 36 be taken. Take the values from the an| 
table opposite 20 and of value below and above 12.1261. Call the 
latter ai, a 2 and a 3 , and arrange them and difference to second differences 
as follows: 

ai 12. 4622 -.9923 .1164 
a 2 11. 4699 -.8759 
a 3 10.5940 

Here A is the first, and A 2 the second, difference of ai and a, a 2 and 
a 3 , it will be noticed are rates 5, 6 and 7 per cent, respectively, and 
hence, h = .01. Inserting these values in the formula, we have: 



(.9923 -.0582 .0582 \-» 
+ ) 
12.1261-12.4622 -9923-0582/ 



.01 

= =.003257. 

3.0706 



And p + (j= 5) = .053257. 



42 FINANCE AND LIFE INSURANCE. 

A closer approximation could be had by taking ai, a 2 , and a 3 from a 
table in which h is x / 2 or a smaller fraction. 

38. In cases where a value is desired which is not included in the 
tables, it may sometimes be conveniently found by interpolation, from 
the tabulated values. Examples of this method are given at Articles 
4 arid 8 of Chapter 3 on Series. Some further elucidation will be given 
here. 

39. Given the amounts of 1 at compound interest for 20 years, as 
follows: At 4%, 2.1911; at 5%, 2.6533; at 6%, 3.2071 and at 7%, 
.86973, to find the compound amount of 1 at 5 l / 2 % for the same term. 

1. Arrange the given values and differences thus: 

A A 2 A 3 

ai =2. 1911 + . 4622 + . 0916+ . 0172. 
a 2 =2 . 6533 +5538 + . 1088. 
a 3 =3.2071 +6626 

a 4 =3.8697. Then we have: 

n = 6 / 2 , A = .4622, A 2 = . 0916 and A 3 = .0172. 

Restating the formula for the nth term: 

(n-1) (n-2) (n-1) (n-2) (n-3) 
an! = a + (n-l) A + A 2 + A 3 +. 

I? L 3 

Substituting the values found, in this equation, we have: 

an~l=2.1911 + ( 3 /2X.4622)+( 3 / 2 X 1 /2X .i/ 2X .0916) + 

1 

(»/« X x k X - V> X X . 0172) = 2. 1911 + . 6933 + .0344 - .0011 = 

2 3 

2.9177, which is a very close approximation, the value being usually 
written, 2.9178. 

40. In further elucidation of the principles discussed in this and 
the preceding chapter and for convenience of reference a series of 
examples with rules and solutions will now be given. 

41. To compute the compound interest on a given principal. 
Rule: Multiply the principal by the rate per cent, expressed decimally. 
Add the product to the principal and take the sum for a new principal 
and proceed as before. Repeat these operations to the end of the 
term. Subtract the principal from the last sum found and the remainder 
will the the compound interest. Example: Compute the compound 
interest for three years at 5%, on 1000. 

1000 X.05=50 and 
10C0 +50 = 1050 . 1050 X . 05 = 52 . 50 and 

1050+52.50 = 1102.50 1102.50 X .05 =55. 125 and 

1 102 . 50 +55 . 125 = 1 157 . 625 and 1 157 . 625 - 1000 = 157 . 625 
which is the compound interest. Answer, $157.63. 



INTEREST— EXAMPLES AND RULES. 43 

42. To compute compound interest by the use of logarithms . 

Rule: First find the logarithm of 1 plus the rate per cent expressed 
decimally. Multiply this value by the term and to the product add 
the log of the principal. The sum will be the log of the compound 
amount, and its anti log the compound amount. Deduct the principal 
from the latter and the remainder will be the compound interest. 

Example: Compute by logs the compound interest on $9675.00 
for 19 years at 472%, the interest to be compounded semi-annually. 

Here there are 38 half-year periods or rests in the payments of 
interest and the rate for each period is one-half of 472% or 2 l U%. 
1 plus this rate is 1 .0225 and its log is .009664. And, .009664x38 = 
.367213. The log of 9675 is 3.985651. Also .367213+3.985651 = 
4.352864=22536.34 and subtracting 9675, we have S12,860.34, 
Answer. 

43. To compute compound interest at the nominal rate. 

Suppose that in the last problem the condition is imposed that the 
interest earned shall be only 472% per annum. That is, that the 
effective rate shall be 472% per annum. The computer must first 
find the nominal rate payable semi-annually which will produce 472% 
per annum. This we have seen, Chapter 5, Article 16, is 2>/ 1.045 — 1. 
That is twice the square root of 1, plus the rate, — 1, which in this 
case is .044505. Computing by the latter rate, we have log 1 .0222525 
= . 009558, which multiplied by 38 gives . 363208. And . 363208 + 
3.985651=4.348859 which is 22328.46. Subtracting the principal 
9675, we have $12653.46, Answer. 

44. To find the time in which a given principal will amount- 
to a given sum at compound interest. 

Rule: When tables of compound amounts are available, the term 
may be found approximately by dividing the given amount by the 
given principal. The quotient will be the amount of 1. Then follow 
down the line of the table under the given rate until the said amount 
of 1 is found. The term will be found under n in the column at the 
left. 

Example: In what time will $1000 accumulate to $1500.00 at 5% 
compound interest? 

Dividing 1500 by 1000, we have 1 .50000 as the compound amount 
of 1. Looking down the 5% column of the compound amounts, sn| 
table, we find the value 1 .477455 opposite n 8, and 1 .5511328 opposite 
n 9 from which we know that the time is less than 9 years and only 
slightly exceeds 8 years. The term can be still more closely approxi- 
mated by interpolating between these terms and amounts. Thus a 
difference of one year in the term makes a difference of 1 . 551328 — 
1 . 477455 or . 073873 in the amount. Then a difference of 1 . 500000 - 

. 022545 

1 . 477455 or . 022545 makes a difference of or . 3 in the term. 

073873 



44 FINANCE AND LIFE INSURANCE. 

So that the term is approximately 8 . 3 years. 

45. To find the term of a loan by the use of logarithms. 

Rule: Subtract the log of the given principal from the log of the 
given amount, and divide the remainder by the log of 1 plus the given 
rate expressed decimally. 

Example: Find the term of the loan of the preceding example. 

The log of 1500 is 3.176091. 

The log of 1000 is 3.000000 and their difference is . 176091. The 
log of 1.05 is .021189 and . 176091-=- .021189 =8.3577, 8 years, 4 
months, 8 days, Answer. 

46. To find the rate per cent of a loan, the principal, amount 
and term being given. 

Rule: When tables of compound amounts are available for 
minutely divided rates of interest a close approximation may be had 
by dividing the given amount by the given rate, thus finding the amount 
of 1 for the given time. Then looking across the page opposite the 
given term in the n column until an amount corresponding, or nearly 
corresponding to the amount of 1 is found. The rate or approximate 
rate will be found at the top of that column. 

Example: Given $1327.85 as the amount of $1000 in 13 yearst 
what is the rate? 

Dividing the amount by 1000, we have 1.32785 as the amount 
of 1 in 13 years. Running across the page opposite n 13, we find 
1 .293607 under 2% and 1 .335436 under 2 1 ,U from which we know the 
rate is nearly 2 1 / 4 per cent. We might get a closer approximation by 
interpolating between the two rates in the usual way. 

47. To find the rate of a loan by the use of logarithms. 

Rule: From the leg of the given amount subtract the log of the 
given principal; divide the remainder by the given term and subtract 
1 from the anti log of the quotient, the remainder will be the rate. 

Example: Find the rate of the problem of Article 46 by loga- 
rithms. 

The log of 1327.84 is 3. 123147. 

The log of 1000 is 3 . 000C00. 
Their difference is .123147. Dividing the latter by 13, we have 
.009473 = 1.02205. Deducting 1, we have .002205 or 2 x / 5 per cent. 
Answer, 27 5 per cent. 

48. To find the rate which a given note or bond will pay 
when sold at a discount or premium. 

Rule: First find the amount of the bond at maturity at the 
rate given in the bond. Then treating the price paid as the principal 
of the loan, find the rate per cent by the rule of Article 47. 



BOND VALUATION RULES. 45 

Example: A 20 year drainage bond for $1000, bearing six per 
cent interest payable semi-annually sells at .96. What rate per cent 
on the price does it pay the purchaser? 

The amount of the loan for 40 periods is, by the table, 3262.04. 
The investment is 960. The leg of 3262.04 is 3.513489. The log 
of 960 is 2.982271 and their difference is ., r 31218. Dividing by 40 
we get the rate for one period, six months, plus 1. That is .013280 = 
1.031 or .062 for a year. Answer, 6V5 per cent. In the above solu- 
tion the distinction between effective and r pminal rates is disregarded 
and the computation made as 1 hough the nominal rate was three 
per cent for six-month periods which we know is in excess of six per 
cent per annum, but this is in accordance with the general practice 
in the commercial world as we understand it. Also the excess cr 
bonus of 140.00 paid at maturity is treated as interest which is correct 
enough if so understood by the parties to the transaction. 

49. To find the price which'may be paid for a given bond or 
note in order to realize a specified rate of interest. 

Rule: Treat the principal sum as a deferred payment and the 
annual interest prescribed in the bond as an annuity for the term of 
the loan. Then find the present value of each at the rate to be realized 
on the purchase price to be paid. Their sum will be the price to be 
paid. 

Example: What price can be paid for a 20-year city bond for 
$1000.00 bearing 4'/ 2 per cent per annum, to realize 6 per cent on the 
investment? 

The present value of 1000 due in 20 years is 1000 X v 20 -. Taking 
v 20 from the 6^ column of the v n table, we have .311805 and 
.311805 X 1000 = 311 .81, the value of the principal. Also the value 
of an annuity of 1 for 20 years taken from the an| table at 6 per 
cent is 11.47 and 11.47x45 gives 516.15 as the present value of 
the annual interest payment of 45.00. The sum of 311.81 and 
516 . 15 is 827 . 96. Answer $827 . 96. 

The foregoing rule does not appty when the rate to be realized is 
convertible at periods different from those named in the bond. See 
Article 53 for a rule of general application. 

50. To find the amount to which a forborne annuity will 
accumulate in a given time at a given rate of interest. 

Rule: Compute the compound interest on 1 for the given term 
at the given rate of interest. Divide the compound interest found 
by the rate of interest used and multiply the quotient by the anrual 
payment of the annuity. 

Example: What would be the amount of an uncollected pension 
of $288 per annum, payable $72 per quarter, after ten years at 4 1 / 2 
per cent interest per annum? 



46 FINANCE AND LIFE INSURANCE. 

The compound interest on 1 for 40 periods at lVsper cent is 
(1. 01 125) 40 -l = .564376. The nominal rate to produce 47 2 per 
cent per annum is 4(1.045)* -1 = . 04426 and .564376-^.04426 = 
12.751 for the term of ten years. Multiplying this by the payment 
288, we have as the whole accumulation, 3672. Answer, $3672. 

51. To find the payments of a Sinking Fund. That is, to 
find the annual payments which will discharge a given sum 
payable at a future date at a given rate of interest. 

Rule: Divide the interest on 1 for one year by the compound 
interest on 1 for the given term, both at the given rate per cent and 
multiply the quotient by the amount to be paid off, the product will 
be the annual payment. 

Example: What annual payment will be required to pay off 
$1000.00 due in ten years, the payments to bear 4 per cent interest? 

The interest on 1 for one year is .04. The compound interest on 
1 for ten years at 4 per cent is . 423312. And . 04-^ . 423312 = . 094493. 
And . 094493 X 1000 = 94 . 493. Answer, $94 . 493. 

Example 2: What monthly payment will mature a $1000.00 
share of Building and Loan stock in six years at 5 per cent? 

There will be 72 periods at 5 /i2 of 1 per cent which will amount to 
1.349018 when computed on a principal of 1. Of this sum, .349018 
is interest. Dividing .00 5 /i2 by .349018 gives .0119379. And 
.0119379X1000 = 11.9379. Answer, $11.94. 

Rule 2: Multiply the sinking fund as given in the tables by the 
sum to be raised, matured or paid off, the product will be the amount 
of the periodical payment. 

Example 3: What semi-annual deposit must be made to replace 
electric light machinery to cost $12000.00 in ten years, the deposits 
to bear four per cent? 

There are 20 deposits to be made and the sinking fund will there- 
fore be computed at 2 per cent. By the table, we find this to be 
.041157. And .041157x12000=493.884. Answer, $493.88. 

52. To find the time in which a given periodical payment 
will amount to a given sum at a given rate per cent. 

Rule: From the logarithm of the annual payment subtract the 
logarithm of the difference between the annual payment and the annual 
interest on the debt and divide the remainder by the logarithm of 1 
, plus the rate of interest borne by the debt. 

Example: How long a term will be required in which to pay off 
a city debt of $4,000,000.00 if an annual deposit of $400,000.00 is 
placed in the sinking fund, the debt bearing 4 x /2 per cent interest? 

The annual deposit being 400000, the annual interest 180000, 
leaves a net deposit of 220000. The log of 400000 is 5 . 602060. The 
log of 220000 is 5.342423. The log of 1.045 is .019116. Then 



VALUATION— INSTALLMENT LOANS. 47 

5.602060-5.342423 = .259637. And .259637^ .019116 = 13.58. 

Answer, 13 years, 7 months. 

Example 2: In what time will a monthly payment of $10.00 
mature a $1000.00 share of building and loan stock at 6 per cent 
interest? 

Here the monthly interest rate is 6 /i 2 or .005. The term is n. 
Using the formula: 

(l+i) n -l U+i) n -l 
Sn| = we have 10 =1000 and 



(l+i) n -l 

=100 and (1 -f-i)" =100i + l and 



n log (l+i)=log (100i + l) and 

(100i + l) .176091 

n=log ■ — ■ — = =81.3 months 

log(l+i) .002166 

Answer, 6 years, 9 months and 9 days. 

Another method of finding the term would be to look down the Sn | 
column under the rate */a% until a value is found which multiplied 
by 10 will produce 1000. Following this suggestion, we find 99.558046 
opposite 81 and 101.055836 opposite 82, and we know therefore that 
the term is between 81 and 82 months. That is, in 81 periods or 
6 9 /n years, the monthly payments will amount to 995.56. The term 
may be still more closely approximated by interpolating between the 
81 and 82 periods in the way already familiar to the reader. 

53. To find the price at which a bond payable in equal 
periodical installments or otherwise may be purchased so as 
to realize a given rate of interest. 

Rule: First find the present value of the capital at the rate 
intended to be realized on the purchase price. Subtract this value 
from the sum to be repaid. Multiply the remainder by the rate of 
the bond, divide the product by the rate to be realized and add the 
result to the present value first found. The sum will be the purchase 
price sought. 

Example: A 4 per cent bond for $1000.00 payable in ten years 
by equal annual installments is offered. What price can be paid for 
it to realize 6 per cent on the investment? 

The present value of the capital to be included in the installments 

1000XaTo| (1000X7.36) 

is: and at 6 per cent, this is =736.00. 

10 10 



48 FINANCE AND LIFE INSURANCE. 

The present value of the interest included in the installments at 6% 
is (1000-736) and (1000-736) X 4 /e =88. FinaUy, 736+88=824. 
Answer, $824.00. 

Example 2: Public improvement bonds for $36000.00 bearing 

5 per cent interest to be repaid in fifteen equal annual installments 
beginning in six years, are offered. What price can be paid to realize 

6 per cent on the series? 

The problem differs from the first only in that the first payment 
is deferred six years, making it necessary to discount the values five 
years. The present value of the payments of principal is therefore, 
36000 X . 747258 X9 . 712249^ 15 =_17418 . 12, the second and third fac- 
tors being respectively v 5 and al5| taken from the 6 per cent tables 
Then 36000-17418. 12 X 5 /e = 15484.90. And finally, 17418.12 + 
15484 . 90 =32903 . 02. Answer, $32903 . 02 or $913 . 97 per 1000. 

Example 3: Suppose the conditions in example 1 are so changed 
that a premium or bonus of 25 per cent shall be included with the prin- 
cipal. What is the solution? 

Here we may treat the 1250 to be paid on the bond as the principaj 
in which case the rate is 4 / x2 5 or 3 l / 5 per cent instead of 4% as specifie^ 
in the bond. That is, 3Vs% on 1250 is equivalent to 4% on 1000.0 

Solving on this theory, we have : 

1250X7.36 3.2 

=920. Also (1250 -920) X = 176.50. And 920 + 176.50 

10 6 

= 1126.50. Answer, $1126.50. 

The rule given and illustrated in this Article has sometimes been 
expressed as a formula, thus: 

i 

A=K+-(C-K) • (29) 



A. is the value of the bond at the rate to be realized by the pur- 
chaser, K the present value of the capital at the rate to be realized, 
C, the capital to be repaid by the borrower, j, the nominal rate paid 
on the capital and i, the rate to be realized by the purchaser. 

In the last problem A =1126.50, K=920, C=1250, j =3.2 and 
i=6. 

In applying the rule or formula, the only difficulty which will be 
encountered will be found in computing the present value of the capital 
included in the payments, that is, in finding K. This is because of 
the various forms which the contract may assume with respect to the 
payments. But whatever the form, if K be correctly computed and 
inserted in the formula, the result will be the value sought. The 
formula is, therefore, of very general application and may be used in 
valuing all sorts of contracts for the repayment of loans. 



VALUATION— INSTALLMENT LOANS. 49 



Thus to solve the problem of Article 49, we have for K, 1000 Xv 20 
at 6% or 311.81. The value of the interest at 6% is 1000-311.81 X 

4.5 

=516. 14 and the purchase price, A $827.95, as before. 

6 

54. To find the rate which an installment loan will pay 
when purchased at a given price. 

Rule: Compute the present value of the payments of capital 
at a rate which by inspection appears to be near the rate which the 
loan will earn at the purchase price, that is, find K of formula (29). 
Subtract this value from the capital to be repaid by the borrower, 
that is, from C. Divide this remainder by the difference between the 
value first found and the purchase money, A, and multiply the quotient 
by the rate of the loan. The result will be the approximate rate of 
interest earned on the investment. In the notation of Article 53, 
this rule may be expressed as follows : 

i(C-K) (30) 



A-K 



If the approximation is not close enough, use the approximate 
rate first found in again computing K and solve for i as before. 

Example: A railroad improvement loan of $800000.00 is author- 
ized, payable in ten years at 5 per cent by equal semi-annual install- 
ments. What rate will be realized by a purchaser who pays .92 for 
the bonds? Cutting off ciphers for convenience, we have C =80, 
A =73.60 and j =2 1 / 2 . Also assume 6 per cent as a trial rate or 3 
per cent semi-annually. By the 3 per cent table, we have a 20J = 

80X14.8775 

14.8775 and =59.51 which is K of the formula. Then 

20 

80-59.51 
X 2 1 / 2 = 1. 454 X. 025 = .03635, which is evidently too 



73.60-59.51 

large. Again finding K by the rate, .03635, we have 80X13.4381 



20 

80-53.7524 

= 53.7524 and X .02V 2 = -033C6 or .0661 per annum. 

73.60-53.7524 

Answer, 6.61 per cent. 

Example 2: What rate will be earned on a six per cent bond for 
$1000.00 payable at par in 20 years, bought at 105? 



v 



50 FINANCE AND LIFE INSURANCE. 

Here, the principal sum is not paid until maturity. The value of 
the principal then is 1000 Xv n at the rate to be realized. Assume 5 
per cent as a trial rate, v 20 at that rate by the table is . 37689. Hence, 
K =376 . 89. Also C = 1000 and A = 1050. Inserting these values in the 
formula, we have: ' 

1000-376.89 

X . 06 = . 05554. 

1050-376.89 

Answer, 5.55 per cent. 

Example 3 : What rate will be earned on a 5 per cent bond payable 
in 16 years interest payable semi-annually, the bond being purchased 
at 95? 

Let it be assumed that the rate earned will be 6 per cent, or 3 per 
cent for half year periods, v 32 at 3 per cent is . 3883 and K is therefoer 
38.83. 

100-38.83 

X .025 = .02723, which double for the yearly rate. 

95-38.83 

Answer, 5.446 per cent. 

The following formula which is said to give a closer approximate 
rate will be given and illustrated : 

(C-A) 

h=I (31) 

C-K 

I, being a trial rate and h, a correction to be added to the rate pres- 
cribed in the bond. 

• Solving the last problem by way of illustration, we have : 
100-95 



X. 03 = .00245. Then 



100-38.83 

( . 025 + .00245) X2 = . 0549. Answer, 5 . 49 per cent. 

5. Suppose a farm loan of $12000, bearing 4 per cent interest, 
the principal of which is payable in three installments of $4000 each, 
at the end of four, eight and twelve years, is offered at $11500, what 
rate of interest will it pay the purchaser? 



VALUATION — INSTALLMENT LOANS. 51 

Let us take a trial rate 4 'A per cent, then by the table the present 
values of the three payments of capital are: 
.838561X4000 = 3354.24 
.703185X4000 = 2812.74 

.589664X4000 = 2358.67 



8525.65 
And by formula 27, and the conditions of the problem, we have 
12000 -1 1500 X. 045 22.5 



0004 and . 04 + . 0C64 = . C464. 



12000-8525.65 3474.35 
Answer, 4.64 per cent. 

If 5 per cent be used as a trial rate, the result is 4 . 66 and we may 
regard 4.65 as a close approximation. 

56. Greater accuracy may be attained by finding two approximate 
rates from two trial rates and interpolating for a new approximate 
rate. For this purpose an excellent formula has been suggested by 
Mr. King. This, with his notation, will be given and illustrated as a 
conclusion to this branch of our subject. Let Ii be the first trial rate, 
I 2 the second trial rate, Ji the first approximate rate, that is, the one 
derived from Ii and J 2 the second approximate rate. Also let A be 
the difference between the trial rates, d the difference between the 
approximate rates, Ji and J 2 and i, the new approximate rate. Then 
we have the formula: 

A(Ji-Ii) (32) 

1=1!+ 

A-d 

57. Drainage district bonds for -$1000 each payable in 15 years 
and bearing 4% interest per annum, are sold at 90. What per cent 
do they pay the purchaser? 

Solution: He gets a bonus of $100 at maturity and 4 per cent 
per annum on $1000. Let us suppose he earns 4V2 per cent, we then 
have Ii = .045 and by formula (31) we have: 

(1000 -900) X. 045 

= . C0934, 

1000-516.72 

so that J, is .04934. Again trying 5% or I 2 = .05, we find J 2 = .04963. 
From which we derive A = . 5, and d = . 029. Inserting the proper 
values in formula (32) we have: 

.5 (.04934 -.04500) 

i=4.5H =4. 5 + .0461 =4.96 per cent. 

.5 -.029 



FINANCE AND LIFE INSURANCE, 



59. A little practice with the aid of the tables given will enable 
the student or investor to make very accurate calculations of the 
values required in the operations of finance. The tables are nearly 
all computed true to six places of decimals giving correct values up to 
100000. If greater accuracy is required, which will rarely occur, the 
short eight figure table of logarithms may be used and the formula 
and rules given will enable the computor to obtain all the values his 
problems will require and as a rule, with little labor. 



CHAPTER VII 
Of The Tables 



There are forty-three tables published in this book which have 
been prepared with the view of enabling the computer to solve with 
little labor nearly all of the problems that arise in transactions re- 
lating to finance and life insurance, after reading the brief texts given 
in the body of the work. 

Table 1. Is a short table of logarithms adapted to computing 
by logarithms to five and to eight places of decimals. Its use is 
explained in Chapter 2. 

Table 2. Gives the amount of one at compound interest at rates 
of interest ranging from 1 / i of one per cent to ten per cent and for 
periods of from one to sixty. By deducting one from these amounts, 
the compound interest is obtained. 

Table 3. Gives the present value of one, discounted at compound 
interest for the same periods and at the same rates included in Table 
2. It is of frequent application in computations relating to all of the 
subjects treated in this book. Its symbol is v n . The superscript, 
n, relating to the number of years or periods of discount, v deducted 
from 1 gives the discount of one for one period. Its symbol is d. 

Table 4. Gives the amount of an annuity of one computed at 
rates of interest from one-fourth of one per cent to ten per cent for 
periods of from one to sixty. Its symbol is Snj, n referring to the 
number of periods in the term. 

Table 5. Is an annuity table. It gives the present value of an 
annuity certain of a payment of one computed for the terms and rates 
of compound interest mentioned in the three preceding tables. It 
and Table 3 are used in computing the values of bonds and loans and 
is otherwise of frequent application. Its symbol is an|. 

Table 6. Gives sinking funds accumulating at the rates of interest 
and for the terms mentioned in the last mentioned table. By dividing 
the sum to be discharged by the sinking fund for the term and rate to 
be realized on the deposits, we get at once the amount of the periodical 
deposits to be made. By adding the rate of the interest to the sinking 
fund, the annuity which a unit will purchase is obtained. 

Tables 7 to 11, inclusive, present the five mortality tables winch 
may be regarded as standard tables in this country. These tables 
give the number living of the assumed group of lives for each year of 
the human life, the number of the group dying each year, the yearly 
probability of living and of dying. They are called life tables and 
are the basis upon which all forms of life insurances and life annuities 
are computed. The American Experience table is the standard table 
for life insurance valuations in probably more than one-half the states 



54 FINANCE AND LIFE INSURANCE. 

in the Union. The Actuaries or Combined Experience Table — also 
known as the Seventeen Office table, is standard in the rest of the 
states. The Carlisle table is employed in many states in computing 
annuities for the valuation of life estates, life incomes and reversions. 
The National Fraternal Congress table is used in Fraternal insurance 
circles in computing rates of contribution or premiums, valuations, 
etc. All four of the tables have had frequent legislative and judicial 
recognition thru out the country and may be relied on as fairly repre- 
senting the rate of mortality. The Northampton table was formerly 
much used in computing annuities and may still be recognized as 
authoritative in some of the courts. 

Tables 12 to 15, inclusive, give commutation columns for tlr 9 
computation of annuities on single lives at the rates of five and six 
per cent according to the Actuaries, Carlisle, Northampton and 
American Experience Tables of Mortality. 

Table 16. Gives joint life commutation columns on two and three 
lives of equal ages at five per cent by the regraduated Carlisle table 
by means of which joint and contingent life annuities on lives of 
different ages may be readily computed. 

Table 17. Differs from 16 only in that it is computed at 6 per 
cent. It being necessary to use the one based on the established rate 
in the state where the valuation is required. 

Tables 18, 19 and 20. are based on the regraduated American 
Experience table at rates of 3 l U, 5 and 6 per cent. The latter two are 
adapted to the computing of joint and contingent life annuities on 
two and three lives in the same manner as the Carlisle table. The 
3V2 per cent columns may be used to compute annuities from which 
joint life insurances may be derived. ' 

Tables 21 to 26, inclusive, are based upon the regraduated Actuaries 
table and may be used in computing joint life annuities at 4, 5 and 6 
per cent on two and three lives. The four per cent table is also adapted 
to computing annuities from which joint life insurances may be derived. 
One or the other of the foregoing tables may be used for valuing an- 
nuities, incomes, reversions and insurances where joint lives are 
involved in all the states. 

Table 27. Gives the force of mortality by the three regraded 
tables last mentioned and is used in connection with those tables in 
making calculations as explained in the text. 

Table 28 to 30. Give joint life annuities at the rates and by the 
tables mentioned in Articles 16 to 26. 

Table 31. Gives the expectation of life as derived from the North- 
ampton, Carlisle, Actuaries and American Experience tables of mor- 
tality. They represent the average number of years lived by a life 
of the ages given. They do not furnish a proper basis for computing 
insurance or annuity values, but are sometimes used in courts where 
values depending upon the duration of human life are involved. 



EXPLANATION OF TABLES. 55 

Table 31, to 39, inclusive, are Commutation tables based on the 
Actuaries, American Experience and Fraternal CongTess tables at 
from 3 to 4 l /s per cent interest. They are computed at rates appro- 
priate for the computation of insurance values. 

Tables 40 to 42, inclusive, give annuities on single lives at from 3 
to 6 per cent based on Actuaries, American Experience and Carlisle 
Mortality Tables. 

Table 43. Gives the net single premiums for an insurance of 1 at 
the rates of 3, 3 l / 2 , 4 and 4Va per cent on the American Experience 
table and at the rates 3, 3V2 and 4 per cent on the Actuaries or Com- 
bined Experience table. They are only given, on ages from 10 to 70 
years, for the reason that little occasion will, in practice, be found for 
the values above that age and space is thereby saved. These premiums 
may be used in computing annual premiums for whole life or limited 
payment policies and in valuations. Single premiums on the Carlisle 
and Northampton tables are omitted for the reason that they are no 
longer used, to any extent, as standards for life insurance contracts. 



56 



FINANCE AND LIFE INSURANCE. 



TABLE NO. la 
Logarithms to Eight' Place's 



N 





1 


2 


3 


4 


1.0 
1.1 
1.2 
1.3 
1.4 


.00000000 
.04139269 
.07918125 
.11394335 
.14612804 


.00432137 
.04532298 
.08278537 
.11727130 
.14921911 


.00860172 
.04928023 
.08635983 
.120573^8 
.15228834 


.01283722 
.05307844 
.08990511 
.12385164 
.15533604 


.01703334 
.05690485 k 
.09342169 
.12710480 
.15836249 


1.5 
1.6 
1.7 

1.8 
1.9 


.17609126 
.20411998 
.23044892 
.25527251 
.27875360 


.17897695 
.20682588 
.23299611 
.25767857 
.28103337 


.18184359 
.20951501 
.23552845 
.26007139 
.28330123 


.18469143 
.21218760 
.23804610 
.26245109 
.28555831 


.18752072 
.21484385 
.24054925 
.26481782 
.28780173 


2.0 
2.1 
2.2 
2.3 

2.4 


.30103000 
.32221929 
.34242268 
.36172784 
.38021124 


.30319606 
.32428246 
.34439227 
.36361198 
.38201704 


.30535137 
.32633586 
.34635297 
.36548798 
.38381537 


.30749604 
.32837960 
.34830486 
.36735592 
.38560627 


.30963017 
.33041377 
.35024802 
.36921586 
.38738983 


2.5 
2.6 
2.7 
2.8 
2.9 


.39794001 
.41497335 
.43136376 
.44715803 
.46239800 


.39967372 
.41664051 
.43296929 
.44870632 
.46389299 


.40140054 
.41830129 
.43456890 
.45024911 
.46538285 


.40312052 
.41995575 
.43616265 
.45178644 
.46686762 


.40483372 
.42160393 
.43775056 
.45331834 
.46834733 


3.0 
3.1 
3.2 
3.3 
3.4 


.47712125 
.49136169 
.50514998 
.51851394 
.53147892 


.47856650 
.49276039 
.50650503 
.51982799 
.53275438 


.48000694 
.49415459 
.50785587 
.52113808 
.53402611 


.48144263 
.49554434 
.50920252 
.52244423 
.53529412 


.48287358 
.49692965 
.51054501 
.52374647 
.53655844 


3.5 

3.6 
3.7 
3.8 
3.9 


.54406804 
.55630250 
.56820172 
.57978360 
.59106461 


.54530712 
.55750720 
.56937391 
.58092498 
.59217676 


.54654266 
.55870857 
.57054294 
.58206336 
.59328607 


.54777471 
.55990663 
.57170883 
.58319877 
.59439255 


.54900326 
.56110138 
.57287160 
.58433122 
.59549622 


4.0 
4.1 
4.2 
4.3 
4.4 


.60205999 
.61278386 
.62324929 
.63346846 
.64345268 


.60314437 
.61384182 
.62428210 
.63447727 
.64443859 


.60422605 
.61489722 
.62531245 
.63548375 
.64542227 


.60530505 
.61595005 
.62634037 
.63648789 
.64640373 


.60638137 
.61700034 
.62736586 
.63748973 
.64738297 


4.5 
4.6 
4.7 
4.8 
4.9 


.65321251 
.66275783 
.67209786 
.68124124 
.69019608 


.65417654 
.66370093 
.67302091 
.68214508 
.69108149 


.65513843 
.66464198 
.67394200 
.68304704 
.69196510 


.65609820 
.66558099 
.67486114 
.68394713 
.69284692 


.65705585 
.66651798 
.67577834 
.68484536 
.69372695 


5.0 
5.1 
5.2 
5.3 
5.4 


.69897000 
.70757018 
.71600334 
.72427587 
.73239376 


.69983773 
.70842090 
.71683772 
.72509452 
.73319727 


.70070372 
.70926996 
.71767050 
.72591163 
.73399929 


.70156799 
.71011737 
.71850169 
.72672721 
.73479983 


.70243054 
.71096312 
.71933129 
.72754126 
.73559890 



FACTOR LOGARITHMS. 



57 



TABLE NO. la. 
Logarithms to Eight Places 



N 


5 


6 


7 


8 


9 


1.0 
1.1 
1.2 
1.3 

1.4 


.02118930 
.06069784 
.09691001 
.13033377 
.16136800 


.02530581 
.06445799 
.10037055 
.13353891 
.16435286 


.02938378 
.06818586 
.10380372 
.13672057 
.16731733 


.03342376 
.07188201 
.10720997 
.13987909 
.17026172 


.03742650 
.07554696 
.11058971 
.14301480 
.17318627 


1.5 

1.6 
1.7 
1.8 
1.9 


.19033170 
.21748394 
.24303805 
.26717173 
.29003461 


.19312460 
.22010809 
.24551267 
.26951294 
.29225607 


.19589965 
.22271647 
.24797327 
.27184161 
.29446623 


.19865709 
.22530928 
.25042000 
.27415785 
.29666519 


.20139712 
.22788670 
.25285303 
.29646180 
.29885308 


2.0 
2.1 
2.2 
2.3 
2.4 


.31175386 
.33243846 
.35218252 
.37106786 
.38916608 


.31386722 
.33445375 
.35410844 
.37291200 
.39093511 


.31597035 
.33645973 
.35602586 
.37474835 
.39269695 


.31800633 
.33845649 
.35793485 
.37657696 
.39445168 


.32014629 
.34044411 
.35983548 
.37839790 
.39619935 


2.5 
2.6 
2.7 
2.8 
2.9 


.40654018 
.42324587 
.43933269 
.45484486 
.46982202 


.40823997 
.42488164 
.44090908 
.45636603 
.47129171 


.40993312 
.42651126 
.44247977 
.45788190 
.47275645 


.41161971 
.42813479 
.44404480 
.45939249 
.47421626 


.41329976 
.42975228 
.44560420 
.46089784 
.47567119 


3.0 
3.1 
3.2 
3.3 
3.4 


.48429984 
.49831055 
.51188336 
.52504481 
.53781910 


.48572143 
.49968708 
.51321760 
.52633928 
.53907610 


.48713838 
.50105926 
.51454775 
.52762990 
.54032947 


.48855072 
.50242712 
.51587384 
.52891670 
.54157924 


.48995848 
.50379068 
.51719590 
.53019970 
.54282543 


3.5 
3.6 
3.7 
3.8 
3.9 


.55022835 
.56229286 
.57403127 
.58546073 
.59659710 


.55145000 
.56348109 
.57518784 
.58658730 
.59769519 


.55266822 
.56466606 
.57634135 
.58771097 
.59879051 


.55388303 
.56584782 
.57749180 
.58883173 
.59988307 


.55509445 
.56702637 
.57863921 
.58994960 
.60097290 


4.0 
4.1 
4.2 
4.3 
4.4 


.60745502 
.61804810 
.62838893 
.63848926 
.64836001 


.60852603 
.61909333 
.62940960 
.63948649 
.64933486 


.60959441 
.62013606 
.63042788 
.64048144 
.65030752 


.61066016 
.62117628 
.63144377 
.64147411 
.65127801 


.61172331 
.62221402 
.63245729 
.64246452 
.65224634 


4.5 
4.6 

4.7 
4.8 
4.9 


.65801140 
.66745295 
.67669361 
.68574174 
.69460520 


.65896484 
.66838592 
.67760696 
.68663627 
.69548168 


.65991620 
.66931688 
.67851838 
.68752896 
.69635639 


.66086548 
.67024585 
.67942790 
.68841982 
.69722934 


.66181269 
.67117284 
.68033551 
.68930886 
.69810055 


5.0 
5.1 
5.2 
5.3 
5.4 


.70329138 
.71180723 
.72015930 
.72835378 
.73639650 


.70415052 
.71264970 
.72098574 
.72916479 1 
.73719264 | 


.70500796 
.71349054 
.72181062 
.72997429 
.73798733 


.70586371 
.71432976 
.72263392 
.73078228 
.73878056 


.70671778 
.71516736 
.72345567 
.73158876 
.73957234 



58 



FINANCE AND LIFE INSURANCE. 



TABLE NO. la. 

Logarithms to Eight Places 



N 





1 


2 


3 


4 


5.5 
5.6 
5.7 

5.8 
5.9 


.74032690 
.74818803 
.75587486 
.76342799 
.77085201 


.74115160 
.74896286 
.75663611 
.76417613 

.77158748 


.74193908 
.74973632 
.75739603 
.76492298 
.77232171 


.74272513 
.75050839 
.75815462 
.76566855 
.77305469 


.74350976 
.75127910 
.75891189 
.76641285 

.77378645 


6.0 
6.1 
6.2 
6.3 
6.4 


.77815125 
.78532984 
.79239169 
.79934055 
.80617997 


.77887447 
.78604121 
.79309160 
.80002936 
.80685803 


.77959649 
.78675142 
.79379038 
.80071708 
.80753503 


.78031731 

.78746047 
.79448805 
.80140371 
.80821097 


.78103694 
.78816837 
.79518459 
.80208926 
.80888587 


6.5 
6.6 
6.7 
6.8 
6.9 


.81291336 
.81954394 
.82607480 
.83250891 
.83884909 


.81358099 
.82020146 
.82672252 
.83314711 
.83947805 


.81424760 
.82085799 
.82736927 
.83378437 
.84010609 


.81491318 
.82151353 

.82801506 
.83442070 
.84073323 


.81557775 
.82216808 
.82865990 
.83505610 
.84135947 


7.0 
7.1 
7.2 
7.3 

7.4 


.84509804 
.85125835 
.85733250 
.86332286 
.86923172 


.84571802 
.85186960 
.85793526 
.86391738 
.86981821 


.84633711 
.85247999 
.85853720 
.86451108 
.87040391 


.84695532 
.85308953 
.85913830 
.86510397 
.87098881 


.84757266 
.85369821 
.85973857 
.86569605 
.87157294 


7.5 
7.6 

7.7 
7.8 
7.9 


.87506126 
.88081359 
.88649073 
.89209460 
.89762709 


.87563994 
.88138466 
.88705438 
.89265103 
.89817648 


.87621784 
.88195497 
.88761730 
.89320675 
.89872518 


.87679498 
.88252454 
.88817949 
.89376176 
.89927319 


.87737135 
.88309336 
.88874096 
.89431606 
.89982050 


8.0 
8.1 
8.2 
8.3 
8.4 


.90308999 
.90848502 
.91381385 
.91907809 
.92427929 


.90363252 
.90902085 
.91434316 
.91960102 
.92479600 


.90417437 
.90955603 
.91487182 
.92012333 
.92531209 


.90471555 
.91009055 
.91539984 
.92064500 

.92582757 


.90525605 
.91062440 
.91592721 
.92116605 
.92634245 


8.5 
8.6 
8.7 
8.8 
8.9 


.92941893 
.93449845 
.93951925 
.94448267 
.94939001 


.92992956 | .93043959 
.93500315 .93550727 
.94001816 1 .94051648 
.94497591 .94546859 
.94987770 | .95036485 


.93094903 
.93601080 
.94101424 
.94596070 
.95085146 


.93145787 
.93651374 
.94151143 
.94645227 
.95133752 


9.0 
9.1 
9.2 
9.3 
9.4 


.95424251 | .95472479 1 .95520654 | .95568775 
.95904139 | .95951SS8 | .95999484 | .96047078 
.96378783 1 .96425963 | .96473092 | .96520170 
.96848295 | .96894968 1 .96941591 | .96988164 
.97312785 | .97358962 | .97405090 | .97451169 


.95616843 
.96094620 
.96567197 
.97034688 
.97497199 


9.5 
9.6 
9.7 
9.8 
9.9 


.97772361 
.98227123 
.98677173 
.99122608 
.99563519 


.97818052 
.98272339 
.98721923 
.99166901 
.99607365 


.97863695 
.98317507 
.98766626 
.99211149 
.99651167 


.97909290 
.98362629 
.98S11284 
.99255352 
.99694925 


.97954837 
.98407703 
.98855896 
.99299510 
.99738638 



FACTOR LOGARITHMS. 



50 



TABLE NO. la. 
Logarithms to Eight Places 





5 


6 


7 


8 


9 


5.5 


.74429298 


.74507479 


j .74585520 


.74663420 


1 .74741181 


5.6 


.75204845 


.75281643 


1 .75358306 


.75434834 


j .75511227 


5.7 


.75966784 


.76042248 


) .76117581 


.76192784 


j .76267856 


5.8 


.76715587 


.76789762 


1 .76863810 


.76937733 


.77011529 


5.9 


.77451697 


.77524626 
.78247262 


.77597433 


.77670118 


( .77742682 


6.0 


.78175375 


.78318869 


.78390358 


.78461729 


6.1 


.78887512 


.78958071 


.79028516 


.79098848 


.79169065 


6.2 


.79588002 


.79657433 


.79726754 


.79795964 


.79865065 


6.3 


.80277337 


.80345712 


.80413943 


.80482068 


.80550086 


6.4 


.80955971 


.81023252 


.81090428 


.81157501 


.81224470 


6.5 


.81624130 


.81690383 


.81756537 


.81822589 


.81888541 


6.6 


.82282165 


.82347423 


.82412583 


.82477646 


.82542612 


6.7 


.82930377 


.82994670 


.83058867 


.83122969 


.83186977 


6.8 


.83569057 


.83632412 


.83695674 


.83758844 


.83821922 


6.9 


.84198480 


.84260924 


.84323278 


.84385542 


.84447718 


7.0 


.84818912 


.84880470 


.84941941 


.85003326 


.85064624 


7.1 


.85430604 


.85491302 


.85551916 


.85612444 


.85672889 


7.2 


.86033801 


.86093662 


.86153441 


.86213138 


.86272753 


7.3 


.86628734 


.86687781 


.86746749 


.86805636 


.86864444 


7.4 


.87215627 


.87273883 


.87332061 


.87390160 


.87448182 


7.5 


.87794695 


.87852180 


.87909588 


.87966921 


.88024178 


7.6 


.88366144 


.88422877 


.88479536 


.88536122 


.88592634 


7.7 


.88930170 


.88986172 


.89042102 


.89097960 


.89153746 


7.8 


.89486966 


.89542255 


.89597473 


.89652622 


.89707700 


7.9 


.90036713 


.90091307 


.90145832 


.90200289 


.90254678 


8.0 


.90579588 


.90633504 


.90687353 


.90741136 


.90794852 


8.1 


.91115761 


.91169016 


.91222206 


.91275330 


.91328390 


8.2 


.91645395 


.91698005 


.91750551 


.91803034 


.91855453 


8.3 


.92168648 


.92220628 


.92272546 


.92324402 


.92376196 


8.4 


.92685671 


.92737036 


.92788341 | 


.92839585 


.92890769 


8.5 


.93196611 


.93247376 


.93298082 


.93348729 


93399316 


8.6 


.93701611 


.93751789 


.93801910 


.93851973 


.93901978 


8.7 


.94200805 


.94250411 


.94299959 


.94349452 


.94398887 


8.8 


.94694327 


.94743372 


.94792362 


.94841297 


.94890176 


8.9 


.95182304 


.95230801 


.95279244 


.95327634 


.95375969 


9.0 


.95664858 


.95712820 


.95760728 


.95808585 


.95856388 


9.1 


.96142109 


.96189547 


.96236934 J 


.96284268 


.96331551 


9.2 


.96614173 ! 


.96661099 


.96707973 j 


|96754798 


.96801571 


9.3 


.97081162 


.971275S5 


.97173959 | 


.97220284 


-97266559 


9.4 


.97543181 


.97589114 


.97634998 j 


.97680834 


.97726621 


9.5 


.98000337 


.98045789 


.98091194 j 


.98136551 


.98181861 


9.6 


.98452731 1 


.98497713 


.9S542647 


.98587536 


.98632378 


9.7 


.98900462 


.98944982 


.98989456 | 


.99033885 


.99078269 


9.8 


.99343623 


.99387691 


.99431715 


.99475694 


.99519629 


9.9 | 


.99782308 1 


.99825934 | 


.99869516 | 


.99913054 | 


.99956549 



60 



FINANCE AND LIFE INSURANCE. 



TABLE NO. lb. 
Logarithms to Eight Places 



N. 







1 


2 


3 


4 


5 


6 


7 


8 


9 


1.0000 
1.0001 
1.0002 
1.0003 
1.0004 


0.0000 
.0001 


0000 
4343 
8685 
3227 
7368 


0434 

4777 
9119 
3461 
7802 


0869 
5211 
9553 
3895 
8237 


1303 
5645 
9988 
4329 
8671 


1737 
6080 
0422 
4764 
9105 


2171 
6514 
0856 
5198 
9539 


2606 
6948 
1290 
5632 
9973 


3040 
7382 
1724 
6066 
0407 


3474 

7817 
2159 
6500 
0841 


3908 
8251 
2593 
6934 
1275 


1.0005 
1.0006 
1.0007 
1.0008 
1.0009 
1.0010 
1.0011 


.0002 
.0003 

.0004 


1709 
6050 
0390 
4730 
9069 
3408 
7746 


2143 

6484 
0824 
5164 
9503 
3842 
8180 


2577 
6918 
1258 
5598 
9937 
4275 
8614 


3012 
7352 
1692 
6031 
0371 
4709 
9048 


3446 
7786 
2126 
6465 
0805 
5143 
9481 


3880 
8220 
2560 
6899 

|1228 
5577 
9915 


4314 
8654 
2994 
7333 
1672 
6011 
0349 


4748 
9088 
3428 
7767 
2106 
6445 
0783 


5182 
9522 
3862 
8201 
2540 
6878 
1217 


5616 
9956 
4296 
8635 
2974 
7312 
1650 


1.0012 
1.0013 
1.0014 
10015 
1.0016 


.0005 
.0006 


2084 
6422 
0759 
5095 
9432 


2518 
6855 
1192 
5529 
9865 


2952 

7289 
1626 
5963 
0299 


3385 
7723 
2060 
6396 
0732 


3819 
8187 
2493 
6830 
1166 


4253 
8590 
2927 
7264 
1600 


4687 
9024 
3361 

7697 
2033 


5120 
9458 
3794 
8131 
2467 


5554 
9891 
4228 
8564 
2900 


5988 
0325 
4662 
8998 
3334 


1.0017 
1.0018 
1.0019 
1.0020 
1.0021 


.0007 
.0008 
.0009 


3767 
8103 
2438 
6772 
1106 


4201 
8536 
2871 
7206 
1540 


4634 
8970 
3305 
7639 
1937 


5068 
9403 
3738 
8022 
2406 


5502 
9837 
4172 
8506 

2840 


5935 
0270 
4605 
8930 
3273 


6369 
0704 
5038 
9373 
3706 


6802 
1137 
5472 
9806 
4140 


7236 
1571 
5905 
0239 
4573 


7669 
2004 
6339 
0673 
5006 


1.0022 
1.0023 
1.0024 
1.0025 
1.0026 


.0010 
.0011 


5440 
9773 
4106 
8438 
2770 


5873 
0206 
4539 
8871 
3203 


6307 
0640 
4972 
9305 
3636 


6740 
1073 
5406 
9738 
4070 


7173 
1506 
5839 
0171 
4503 


7606 
1939 
6272 
0604 
4936 


8040 
2373 
6705 
1037 
5369 


8473 
2806 
7138 
1471 
5802 


8906 
3239 

7572 
1904 
6235 


9340 
3673 
8005 
2337 
6668 


1.0027 
1.0028 
1.0029 
1.0030 
1.0031 


.0012 
.0013 


7101 
1433 
5763 
0093 
4423 


7535 
1866 
6196 
0526 
4856 


7968 
2299 
6629 

0959 
5289 


8401 
2732 
7062 
1392 

5722 


8834 
3165 
7495 
1825 
6155 


9267 
3598 
7928 
2258 
6588 


9700 
4031 
8361 
2691 
7021 


0133 
4464 
8794 
3124 
7454 


0566 

4897 
9227 
3557 

7887 


0999 
5330 
9660 
3990 
8319 


1.0032 
1.0033 
1.0034 
1.0035 
1.0036 


.0014 
.0015 


8752 
3081 
7410 
1738 
6065 


9185 
3514 

7842 
2170 
6498 


9618 
3947 
8275 
2603 
6931 


0051 
4380 
8708 
3036 
7363 


0484 
4813 
9141 
3469 
7796 


0917 
5246 
9574 
3902 
8229 


1350 
5678 
0007 
4334 
8662 


1783 
6111 
0439 
4767 
9094 


2215 
6544 

0872 
5200 
9527 


2648 
6977 
1305 
5633 
9960 


1.0037 
1.0038 
1.0039 
1.0040 
1.0041 


.0016 
.0017 


0392 
4719 
9045 
3371 
7697 


0825 
5152 
9478 
3804 
8129 


1258 
5584 
9911 
4236 
8562 


1690 
6017 
0343 
4669 
8994 


2123 
6450 
0776 
5102 
9427 


2556 
6832 
1208 
5534 
9859 


2988 
7315 
1641 
5967 
0292 


3421| 3854 
7748 8180 
3074| 2506 
6399' 6832 
0724| 1157 


4286 
8613 
2939 
7264 
1589 


1.0042 
1.0043 
1.0044 
1.0045 
1.0046 
1.0047 
1.0048 
1.0049 


.0018 
.0019 

.0020 
.0021 


2022 
6346 
0670 
4994 
9317 
3640 
7963 
2285 


2454 
6779 
1103 
5426 
9750 
4072 
8395 
2717 


2887 
7211 
1535 
5859 
0182 
4505 
8827 
3149 


3319 
7644 
1968 
6291 
0614 
4937 
9259 
3581 


3752 
8076 
2400 
6723 
1047 
5369 
9691 
4013 


4184 
8508 
2830 
7156 
1479 
5801 
0124 
4445 


4616 
8941 
3265 
7588 
1911 
6234 
0556 
4878 


5049 
9373 
3697 
8020 
2343 
6666 
0988 
5310 


5481 
9806 
4129 
8453 
2776 
7098 
1420 
5742 


5914 
0238 
4562 
8885 
3208 
7530 
1852 
6174 



FACTOR LOGARITHMS. 



61 



TABLE NO. lb. 
Logarithms to Eight Places 



X. 







1 


2 


3 


4 


5 


6 


7 8 


9 


1.0050 


.0021 


66061 7038| 7470| 79031 8335| 8767| 9199| 9631! 0063| 0495 


1.0051 


.0022 


0927! 1359J 1791i 2224 


2656 


3088 


3521| 3952{ 4384| 4816 


1.0052 




5248| 5680| 6112 


6544 


6976 


7408 


7840J 82721 87041 9136 


1.0053 




9568| 0000! 0432 


0864 


1296 


1728 


2160! 2592| 3024( 3456 


1.0054 


.0023 


3888 


4320) 4752 


5184 


5616 


6048 


6480( 6912| 7344J 7776 


1.0055 




8207 


86391 9071 


9503 


9935 


0367 


0799 


1231! 1663 


2095 


1.0056 


.0024 


2526 


2958 3390 


3822 


4254 


4686 


5118 


55491 5981 


6413 


1.0057 




6845 


72771 7709 


8140 


8572 


9004 


9436 


9868) 0300 


0713 


1.0058 


.0025 


1163 


1595 2027 


2458 


2890 


3322 


3754 


4186| 4617 


5049 


1.0059 




5481 


5913| 6344| 6776 


7208 


7639 


8071 


8503| 8935 


9366 


1.0060 




9798J 0230) 0661| 1093 


1525 


1957 


2388 


2820J 3252( 3683 


1.0061 


.0026 


41151 4547 


49781 5410 


5842 


6273 


6705 


7136| 7568) 8000 


1.0062 




8431j 8863 


9295 9726 


0158 


0589 


1021 


1453| 1884 2316 


1.0063 


.0027 


27471 3179 


3610! 4042 


4474 


4905 


5337 


5768| 6200| 6631 


1.0064 




7063J 7494| 7926J 8357 


8799 


9220 


9650 


0083| 0515J 0946 


1.0065 


.0028 1 


3104 


3525 


3967 


4398| 4830| 5261 


1.0066 




5693| 61241 6555| 6987 


7418 


7850 


8281J 8713J 9144| 9575 


1.0067 


.0029 


0007! 0438 


0870 


1301 


1732 


2164 


25951 3027| 3458| 3889 


1.0068 




4321 


4752 


5183 


5615 


6046 


6477 


6909| 7340| 77711 8203 


1.0069 




8634 


9065 


9497 


9928 


0359 


0791 


1222 1653| 2084 


2516 


1.0070 


.0030 


2947 


3378 


3810 


4241 


4672 


5103 


5535| 5966| 6397 


6828 


1.0071 




7260 


7691 


8122 


8553 


8984 


9416 


9847 0278| 0709 


1141 


1.0072 


.0031 


1572 


2003 


2434 


2865 


3296 


3728 


4159| 4590| 5021 


5452 


1.0073 


5883 


6315 


6746 


7177 


7608 


i 8039 


8470| 8901| 9332 


9764 


1.0074 


.0032 


0195 


0625 


1057 


1488 


1919 


| 2350 


27811 3212| 3643 


4074 


1.0075 




4505 


4937 


5368 


5799 


6230 


6661 


70921 7523| 7954 


8385 


1.0076 




8816 


9247 


9678 


0109 


0540 


0971 


1402| 1833| 2264 


2695 


1.0077 


.0033 


3126| 3557 


3988 


4419 


4850 


5281 


5712| 6143| 6574 


7004 


1.0078 




7435 7866 


8297 


8728 


9159| 9590 


0021| 0452| 0883 


1314 


1.0079 


.0034 


1745| 2175 


2606 


3037 


3468 


3899 


4330! 4761| 5192 


5622 


1.0080 




6053| 6484 


6915| 7346 


7777 


8207 


8638| 90691 9500 


9931 


1.0081 


0035 i 


2085 


2515 


2946J 3377| 3808] 4239 


1.0082 




4669 5100{ 5531J 5962 


6392 


6823 


7254 


7685! 81151 8546 


1.0083 




8977 9407 9838 0269 


0700 


1130 


1561 


1992] 2422| 2853 


1.0084 


.0036 


3284 3714| 4145| 4576 


5006 


5437 


5868 


6298| 6729| 7160 


1.0085 




7590! 8021| 8452| 8882 


9313 


9743 


0174| 0605| 10351 1466 


1.0086 


.0037! 1896J 2327! 27581 3188 


3619 


4049 


44801 4910| 5341| 5772 


1.0087 


1 62021 6633' 7063] 7494 


7924 


8355 


8785| 9216| 9646| 0077 


1.0088 


.0038 0507; 0938J 1368! 1799 


2229 


2660 


3090| 3521| 3951| 4382 


1.0089 




4812| 5243; 5673| 6104 


6534| 6964 


7395| 7825 8256| 8686 


1.0090 




9117, 9547| 99771 0408 


0838 


1269 


1699| 2129| 2560: 2990 


1.0091 


.0039 


34211 3851| 428l| 4712 


5142 


5572 


6003| 6433! 6864| 7294 


1.0092 




77241 81551 8585! 9015 


9445 


9876 


0306! 0736| 1167! 1597 


1.0093 


.0040 


2027] 2458! 28881 3318 


3748 


4179 


4609| 4639! 5470[ 5900 


1.0094 




6330| 6760| 719l| 7621 


8051 


8481 


8911| 9342| 9772| 0202 


1.0095 


.0041 


0632| 1063| 1493| 1923 


| 2353 


| 2783 


3213| 3644) 4074| 4504 


1.0096 




4934! 5364: 57951 6225 


665C 


7085 


75151 79451 83751 8806 


1.0097 




9236| 9666i 0096) 0526 


0956 


1 1386 


18161 2246| 2676| 3107 


1.0098 


.0042 


3537| 3967! 43971 4827 


5257) 5687 


6117] 65471 6977| 7407 


1.0099 




7837 


8267 


8697 


| 9127 


9557] 9987 


0417 


0847] 1277 


1707 



62 



FINANCE AND LIFE INSURANCE. 



TABLE NO. II. 

The Amount of One Dollar at Compound Interest for the 
Time and at the Rate Per Cent Stated, (i-f-i)n 



Periods 


1-4% 


1-3 % 


3-8% 


5-12 % 


1-2% 


1 


1.002500 


1.003333 


1.003750 


1.004167 


1.005000 


2 


1.005006 


1.006678 


1.007514 


1.008350 


1.010025 


3 


1.007519 


1.010033 


1.011292 


1.012552 


1.015075 


4 


1.010038 


1.013401 


1.015082 


1.016771 


1.020151 


5 


1.012563 


1.016878 


1.018891 


1.021008 


1.025251 


6 


1.015094 


1.020167 


1.022712 


1.025262 


1.030378 


7 


1.017632 


1.023568 


1.026547 


1.029536 


1.035529 


8 


1.020176 


1.026980 


1.030396 


1.033824 


1.040707 


9 


1.022726 


1.030403 


1.034260 


1.038131 


1.045906 


10 


1.025283 


1.033833 


1.038136 


1.042478 


1.051140 


1 


1.027846 


1.037284 


1.042032 


1.046800 


1.056396 


2 


1.030416 


1.040742 


1.045939 


1.051162 


1.061678 


3 


1.032992 


1.044211 


1.049862 


1.055543 


1.066986 


4 


1.035574 


1.047691 


1.053799 


1.059940 


1.072321 


5 


1.038163 


1.051184 


1.057748 


1.064356 


1.077683 


6 


1.040757 


1.054688 


1.061717 


1.0687*89 


1.083071 


7 


1.043361 


1.058205 


1.065698 


1.073244 


1.088487 


8 


1.045969 


1.061731 


1.069695 


1.077716 


1.093928 


9 


1.048584 


1.065268 


1.073706 


1.082267 


1.099398 


20 


1.051206 


1.068821 


1.077733 


1.086716 


1.104896 


1 


1.053834 


1.072383 


1.081774 


1.091244 


1.110420 


2 


1.056468 


•1.075958 


1.085831 


1.095590 


1.115972 


3 


1.059109 


1.079544 


1.089904 


1.100356 


1.121552 


4 


1.061757 


1.083143 


1.093990 


1.104941 


1.127160 


5 


1.064411 


1.086754 


1.098092 


1.109545 


1.132796 


6 


1.067072 


1.090379 


1.102240 


1.114169 


1.138459 


7 


1.069741 


1.094011 


1.106343 


1.118808 


1.144152 


8 


1.072414 


1.097657 


1.110492 


1.123472 


1.149873 


9 


1.075095 


1.101316 


1.114656 


1.128153 


1.155622 


30 


1.077784 


1.104987 


1.118836 


1.132741 


1.161400 


1 


1.080478 


1.108670 


1.123032 


1.137574 


1.167207 


2 


1.083178 


1.112366 


1.127243 


1.142314 


1.173043 


3 


1.085887 


1.116074 


1.131495 


1.147074 


1.178908 


4 


1.088602 


1.119794 


1.135715 


1.151850 


1.184803 


5 


1.091323 


1.123527 


1.139992 


1.156653 1 


1.190727 


6 


1.094051 


1.127272 


1.144248 


1.161472 


1.196681 


7 


1.096787 


1.131029 


1.148538 


1.166312 


1.202664 


8 


1.099529 


1.134800 


1.152846 


1.171171 


1.208677 


9 


1.102277 


1.138582 


1.157168 


1.176051 1 


1.214721 


40 


1.105033 


3.142377 


1.161508 


1.180951 1 


1.220794 


1 


1.107796 


1.146185 


1.165864 


1.185872 


1.226898 


2 


1.110570 


1.150006 


1.170236 


1.1P0813 1 


1.233033 


3 


1.113342 


1.153839 


1.174624 


1.195775 


1.239198 


4 


1.117625 


1.157686 | 


1.179029 | 


1.200757 1 


1.245394 


5 


1.118918 


1.161544 


1.183450 


1.205760 


1.251621 


6 


1.121713 


1.165416 


1.187888 


1.210784 


1.257879 


7 


1.124517 


1.169301 


1.192343 1 


1.215830 ! 


1.264168 


8 


1.127328 


1.173198 I 


1.196814 ' 


1.220895 1 


1.270494 


9 


1.130146 


1.177109 1 


1.201302 


1.225982 1 


1.276847 


50 


1.132972 


1.181036 


1.205807 


1.231090 | 


1.283226 


1 


1.135904 


1.184969 | 


1.210329 1 


1.236220 I 


1.289639 


2 


1.138641 


1.188920 


1.214868 


1.241371 1 


1.296090 


3 


1.141490 


1.192883 


1.219423 


1.246544 


1.302571 


4 


1.144344 


1.196859 


1.223996 


1.251737 1 


1.309084 


5 


1.147205 


1.200849 


1.228586 


1.256953 


1.315629 


6 


1.150073 


1.204851 


1.233194 


1.262190 


1.322207 


7 


1.152948 


1.208868 


1.237818 


1.267450 


1.328818 


8 


1.155830 


1.212894 


1.242460 


1.272730 


1.335462 


9 


1.158720 


1.-216940 


1.247119 


1.278033 


1.342133 


60 


1.161617 


1.220996 


1.251796 


1.283359 


1.348850 



COMPOUND INTEREST. 



TABLE NO. II. Continued. 



The Amount of One Dollar at Compound Interest for the 
Time and at the Rate Per Cent Stated. 



Periods 


1-4% 


1-3 c /o 


3-8 % 


5-12 % 


1-2% 


61 


1.164521 


1.225067 


1.256461 


1.288706 


1.355594 


2 


1.167432 


1.229150 


1.261202 


1.294075 


1.362372 


3 


1.170351 


1.233247 


1.265902 


1.299467 


1.369184 


4 


1.173276 


1 1.237358 


1.270678 


1.304882 


1.376030 


5 


1.176210 


1.241483 


1.275443 


1.310319 


1.382910 


6 


1.179150 


1.245621 


1.280227 


1.315772 


1.389825 


7 


1.182098 


1.249973 


1.285027 


1.321230 


1.396772 


8 


1.185058 


1.253942 


1.289816 


1.326766 


1.403758 


9 


1.188016 


1.258119 


1.294683 


1.332294 


1.410776 


70 


1.190986 


1.262312 


1.299538 


1.337846 


1.417830 


1 


1.193991 


1.266520 


1.304412 


1.343420 


1.424920 


2 


1.196948 


1.270742 


1.309303 


1.349018 


1.432044 


o 


1.199941 


1.274978 


1.314213 


1.354640 


1.439204 


4 


1.202941 


1.279228 


1.319141 


1.360283 


1.446401 


5 


1.205948 


1.283491 


1.324088 


1.365951 


1.453632 


6 


1.208963 


1.287770 


1.329050 


1.371638 


1.460901 


7 


1.211985 


1.292063 


1.334038 


1.377357 


1.468206 


8 


1.215015 


1.296369 


1.339040 


1.383096 


1.475546 


9 


1.218053 


1.300691 


1.344030 


1.388859 


1.482924 


80 


1.221098 


1.305026 


1.349102 


1.394645 


1.490339 


1 


1.224151 


1.309376 


1.354161 


1.400457 


1.497790 


2 


1.227211 


1.313741 


1.359239 


1.406292 


1.505279 


3 


1.230279 


1.318120 


1.364336 


1.412151 


1.512806 


4 


1.233355 


1.322514 


1.369452 


1.418035 


1.520369 


5 


1.236438 


1.326922 


1.374585 


1.423944 


1.527971 


6 


1.239529 


1.331345 


1.379742 


1.429877 


1.535611 


7 


1.242628 


1.335783 


1.384916 


1.435835 


1.543289 


8 


1.245738 


1.340236 


1.390110 


1.441817 


1.551006 


9 


1.248850 


1.344703 


1.395323 


1.447825 


1.558761 


90 


1.251971 


1.349185 


1.400555 


1.453858 


1.566555 


1 


1.255101 


1.353683 


1.405808 


1.459915 


1.574387 


2 


1.258239 


1.358198 


1.411079 


1.465998 


1.582259 


3 


1.261385 


1.362722 


1.416771 


1.472107 


1.590171 


4 


1.264538 


1.367265 


1.421882 


1.478240 


1.598118 


5 


1.267699 


1.371822 


1.427010 


1.484400 


1.606112 


6 


1.270868 


1.376395 


1.432365 


1.490585 


1.614142 


7 


1.274046 


1.380983 


1.437736 


1.496792 


1.622213 


8 


1.277231 


1.385587 


1.443128 


1.503032 


1.630325 


9 


1.280424 


1.390305 


1.448539 


1.509295 


1.638476 


100 


1.283624 


1.394839 


1.453971 


1.515588 


1.646668 


1 


1.286834 


1.399488 | 


1.459424 


1.521899 


1.654902 


2 


1.290051 


1.404153 


1.464894 


1.528237 


1.663176 


3 


1.293276 


1.408834 1 


1.470390 


1.534608 


1.671492 


4 


1.296509 


1.413530 


1.475904 


1.541002 


1.679850 


5 


1.299751 


1.418242 


1.481439 


1.547423 


1.688249 


6 


1.303000 


1.422970 


1.486994 


1.553871 


1.696690 


7 


1.306260 


1.427712 


1.492570 


1.560345 


1.705174 


8 


1.309523 


1.432471 


1.498167 


1.566847 


1.713699 


9 


1.312797 


1.437247 


1.503787 


1.573375 


1.722268 


110 


1.316079 


1.442037 ! 


1.509418 


1.579931 


1.730879 


1 


1.319369 


1.446844 


1.515085 1 


1.586514 


1.739534 


2 


1.322667 


1.451665 1 


1.520767 


1.593122 


1.748232 


3 


1.325974 


1.456502 


1.526469 


1.599862 


1.756973 


4 


1.329289 


1.461361 


1.532193 


1.606428 1 


1.765758 


5 


1.332612 


1.466235 i 


1.537939 1 


1.613121 1 


1.774586 


6 


1.335944 


1.471119 ! 


1.543707 1 


1.619843 


1.783459 


7 


1.339284 


1.476023 


1.549495 ' 


1.626592 1 


1.792377 


8 i 


1.342632 ! 


1.480943 ! 


1.55530*6 | 


1.633370 1 


1.801338 


9 


1.345988 


1.485880 ! 


1.561139 1 


1.640175 


1.810345 


120 


1.349354 


1.490833 | 


1.566992 | 


1.646630 | 


1.819397 



64 



FINANCE AND LIFE INSURANCE. 



TABLE NO. II. Continued. 

The Amount of One Dollar at Compound Interest for the 
Time and at the Rate Per Cent Stated. (l+i) n 



Periods 


5-8% 


3-4% 


7-8% 


1% 


1 1-8 % 


1 


1.006250 


1.007500 


1.008750 


1.010000 


1.011250 


2 


1.012539 


1.015056 


1.017596 


1.020100 


1.022663 


3 


1.018865 


1.022690 


1.026510 


1.030301 


1.034131 


4 


1.025235 


1.030339 


1.035503 


1.040604 


1.045765 


5 


1.031643 


1.038067 


1.044571 


1.051010 


1.057531 


6 


1.038091 


1.045852 


1.053724 


1.061520 


1.069427 


7 


1.044576 


1.043698 


1.062955 


1.072135 


1.081458 


8 


1.051106 


1.061599 


1.072266 


1.082857 


1.093624 


9 


1.057679 


1.069536 


1.081659 


1.093685 


1.105927 


10 


1.064287 


1.077582 


1.091134 


1.104622 


1.118369 


1 


1.070939 


1.085666 


1.100693 


1.115668 


1.130951 


2 


1.077633 


1.093807 


1.110335 


1.126825 


1.143674 


3 


1.084368 


1.102010 


1.120062 


1.138093 


1.156540 


4 


1.091145 


1.110275 


1.129873 


1.149474 


1.169524 


5 


1.097965 


1.118603 


1.139771 


1.160969 


1.182709 


6 


1.104826 


1.126992 


1.149755 


1.172579 


1.196014 


7 


1.111732 


1.135444 


1.159827 


1.184304 


1.209467 


8 


1.118583 


1.143960 


1.169987 


1.196174 


1.223076 


9 


1.125672 


1.152540 


1.180236 


1.208109 


1.236835 


20 


1.132708 


1.161182 


1.190575 


1.220190 


1.250750 


1 


1.139787 


1.169993 


1.201005 


1.232391 


1.264821 


2 


1.146910 


1.178667 


1.211525 


1.244716 


1.274050 


3 


1.154074 


1.187571 


1.222136 


1.257163 


1.293439 


4 


1.161272 


1.196413 


1.232813 


1.269735 


1.307990 


5 


1.168585 


1.205386 


1.243644 


1.282432 


1.322705 


6 


1.175953 


1.214426 


1.254539 


1.295256 


1.337586 


7 


1.183202 


1.223506 


1.265529 


1.308209 


1.352633 


8 


1.190597 


1.232711 


1.277615 


1.321291 


1.317851 


9 


1.198038 


1.241956 


1.287798 


1.334504 


1.383239 


30 


1.205515 


1.251271 


1.299079 


1.347849 


1.398800 


1 


1.213060 


1.260650 


1.310460 


1.361327 


1.414537 


2 


1.220642 


1.270110 


1.321938 


1.374940 


1.430450 


3 


1.228271 


1.279626 


1.333519 


1.388690 


1.446543 


4 


1.235949 


1.289233 


1.345200 


1.402577 


1.462816 


5 


1.243673 


1.298903 


1.356984 


1.416603 


1.479273 


6 


1.251445 


1.308644 


1.368871 


1.430769 


1.495915 


7 


1.259267 


1.318456 


1.380862 


1.445076 


1.512744 


8 


1.267134 


1.328348 


1.392991 


1.459527 


1.529762 


9 


1.275054 


1.338310 


1.405151 


1.474122 


1.546972 


40 


1.283026 


1.348348 


1.417470 


1.488864 


1.564376 


1 


1.291045 


1.358460 


1.427884 


1.503752 


1.581975 


2 


1.299114 


1.368649 


1.442413 


1.518789 


1.599772 


3 


1.307233 


1.378913 


1.455049 


1.533977 


1.617770 


4 


1.315404 


1.389255 


1.467795 


1.549318 


1.635969 


5 


1.323625 


1.399675 


1.480653 


1.564811 


1.654374 


6 


1.331897 


1.410172 


1.493623 


1.580459 


1.672986 


7 


1.340222 


1.420749 


1.506711 


1.596263 


1.691807 


8 


1.348598 


1.431404 


1.519906 


1.612226 


1.710840 


9 


1.357024 


1.442140 


1.533221 


1.628348 


1.730087 


50 


1.365508 


1.452956 


1.546652 


1.644632 


1.749550 


1 


1.374053 


1.463853 


1.559412 


1.661078 


1.769232 


2 


1.382630 


1.474832 


1.573058 


1.677690 


1.789137 


3 


1.391274 


1.485893 


1.586822 


1.694466 


1.809264 


4 


1.399968 


1.497052 


1.600707 


1.711411 


1.829614 


5 


1.408717 


1.508266 


1.614713 


1.728525 


1.850202 


6 


1.417521 


1.519577 


1.628841 


1.745810 


1.871016 


7 


1.426381 


1.530974 


1.643093 


1.763268 


1.892065 


8 


1.435296 


1.542456 


1.657471 


1.780901 


1.913351 


9 


1.444267. 


1.554024 


1.671973 


1.798711 


1.934876 


60 


1.453293 


1.565681 


1.686603 


1.816676 


1.956644 



COMPOUND INTEREST. 



65 



TABLE NO. II. Continued. 

The Amount of One Dollar at Compound Interest for the 
Time and at the Rate Per Cent Stated. (l+i)» 



Periods 


1 1-4 % 


1 3-8 % 


1 1-2 % 


1 5-8 % 1 3-4 % 


1 


1.012500 


1.013750 


1.015000 


1.016250 


1.017500 


2 


1.025156 


1.027688 


1.030225 


1.032764 


1.035306 


3 


1.037971 


1.041819 


1.045678 


1.049547 


1.053424 


4 


1.050945 


1.056144 


1.061368 


1.066601 


1.071859 


5 


1.064082 


1.073135 


1.077284 


1.083934 


1.090617 


6 


1.077383 


1.085388 


1.093443 


1.101548 


1.109702 


7 


1.090951 


1.100312 


1.109845 


1.119446 


1.129122 


8 


1.104486 


1.115442 


1.126493 


1.137639 


1.148881 


9 


1.118292 


1.130779 


1.143390 


1.156120 


1.168987 


10 


1.132271 


1.146322 


1.160541 


1.174913 


1.189444 


1 


1.146424 


1.162089 


1.177949 


1.194005 


1.210260 


2 


1.160754 


1.178068 


1.195618 


1.213406 


1.231439 


3 


1.175264 


1.194264 


1.213552 


1.233125 


1.252989 


4 


1.189954 


1.210687 


1.231756 


1.253164 


1.274917 


5 


1.204829 


1.227335 


1.250232 


1.273527 


1.297228 


6 


1.219889 


1.244210 


1.268986 


1.294223 


1.319929 


7 


1.235137 


1.261318 


1.288020 


1.315254 


1.343028 


8 


1.250577 


1.278661 


1.307340 


1.336630 


1.366531 


9 


1.266209 


1.296243 


1.326951 


1.358347 


1.390445 


20 


1.282037 


1.314066 


1.346855 


1.380423 


1.414778 


1 


1.298062 


1.332135 


1.367058 


1.402851 


1.439537 


2 


1.317318 


1.350452 


1.387564 


1.425648 


1.464728 


3 


1.330716 


1.369020 


1.408377 


1.448815 


1.490361 


4 


1.347351 


1.387844 


1.429503 


1.472358 


1.516443 


5 


1.364193 


1.406927 


1.450945 


1.496284 


1.542980 


6 


1.381245 


1.426272 


1.472709 


1.520599 


1.569983 


7 


1.398511 


1.449217 


1.494800 


1.545308 


1.597457 


8 


1.415992 


1.465765 


1.517222 


1.570420 


1.625412 


9 


1.433692 


1.485919 


1.539980 


1.595938 


1.653857 


30 


1.451613 


1.506350 


1.563080 


1.621872 


1.682800 


1 


1.469758 


1.527063 


1.586526 


1.648228 


1.712249 


2 


1.488130 


1.548060 


1.610324 


1.675012 


1.742213 


3 


1.510205 


1.569346 


1.634479 


1.702231 


1.772702 


4 


1.525566 


1.590924 


1.658996 


1.729892 


1.803724 


5 


1.544636 


1.612800 


1.683881 


1.758003 


1.835289 


6 


1.567549 


1.634973 


1.709139 


1.786571 


1.867407 


7 


1.583492 


1.657456 


1.734777 


1.815602 


1.900087 


8 


1.603286 


1.680246 


1.760798 


1.845106 


1.933338 


9 


1.623328 


1.703351 


1.787210 


1.875087 


1.967172 


40 


1.643619 


1.726771 


1.814018 


1.905555 


2.001597 


1 


1.664164 


1.750514 


1.841229 


1.936525 


2.036625 


2 


1.684967 


1.774583 


1.868847 


1.967993 


2.072266 


3 


1.706029 


1 1.798984 


1.896879 


1.999973 


2.108530 


4 


1.727354 


1 1.823720 


1.925333 


2.032480 


2.145430 


5 


1.748938 


1.848797 


1.954213 


2.065500 


1.182975 


6 


1.770807 


1.874217 


1.983521 


2.099064 


2.221177 


7 


1.792942 


1.899988 


2.013279 


2.133177 


2.260048 


8 


1.815343 


| 1.926113 


2.043478 


2.167839 


2.299598 


9 


1.838046 


1 1.952596 


2.074130 


2.203065 


2.339842 


50 


1.861021 


| 1.979445 


2.105242 


2.238865 


2.380789 


1 


1.884295 


| 2.006663 


2.136821 


2.275247 


2.422452 


2 


1.907838 


1 2.034254 


2.168873 


2.312220 


2.464845 


3 


1.931686 


| 2.062225 


! 2.201406 


2.349793 


2.507980 


4 


1.955833 


| 2.090581 


| 2.234427 


! 2.387977 I 2.551863 


5 


1.980280 


I 2.119327 


2.267943 


2.426782 | 2.596527 


6 


2.005034 


! 2.148467 


! 2.301963 


I 2.466218 | 2.641966 


7 


2.030097 


| 2.178008 


I 2.336492 


| 2.506293 | 2.688200 


8 


2.055472 


| 2.207956 


1 2.371539 


f 2.547022 I 2.735344 


9 


2.081166 


2.238315 


I 2.407112 


I 2.588409 | 2.783110 


60 


2.107181 


1 2.269093 


| 2.443219 


| 2.630385 


| 2.831815 



FINANCE AND LIFE INSURANCE. 



TABLE NO. II. Continued. 



The Amount of One Dollar at Compound Interest for the 
Time and at the Rate Per Cent Stated. (l-fi) n 



Periods 


1 7-8 % 


2% 


2 1-4 % 


2 1-2 % 


2 3-4 % 


1 


1.018750 


1.020000 


1.022500 


1.025000 


1.027500 


2 


1.037851 


1.040400 


1.045506 


1.050625 


1.055756 


3 


1.057311 


1.061208 


1.069030 


1.076891 


1.084789 


4 


1.077136 


1.082422 


1.093833 


1.103813 


1.114621 


5 


1.097332 


1.104081 


1.111678 


1.131308 


1.145273 


6 


1.117967 


1.126162 


1.242825 


1.159693 


1.176768 


7 


1.138867 


1.148686 


1.168539 


1.188686 


1.209129 


8 


1.160221 


1.171659 


1.194831 


1.218403 


1.242380 


9 


1.181775 


1.195093 


1.221714 


1.248863 


1.276546 


10 


1.204137 


1.218994 


1.244920 


1.280085 


1.311651 


1 


1.226715 


1.243374 


1.277310 


1.312087 


1.347721 


2 


1.249715 


1.268242 


1.306050 


1.344889 


1.384784 


3 


1.273148 


1.293607 


1.335436 


1.378511 


1.422865 


4 


1.297019 


1.319479 


1.365483 


1.412974 


1.461994 


5 


1.321338 


1.345868 


1.396207 


1.448298 


1.502199 


6 


1.346113 


1.372786 


1.427621 


1.484506 


1.543509 


7 


1.371353 


1.400241 


1.459743 


1.526618 


1.585956 


8 


1.397283 


1.428246 


1.492587 


1.559659 


1.629569 


9 


1.423261 


1.456811 


1.526170 


1.598650 


1.674383 


20 


1.449946 


1.485947 


1.560509 


1.638616 


1.720428 


1 


1.477133 


1.515666 


1.595621 


1.679582 


1.767740 


2 


1.504829 


1.545980 


1.631522 


1.721571 


1.816353 


3 


1.533044 


1.576899 


1.668231 


1.764611 


1.866303 


4 


1.561790 


1.608437 


1.705766 


1.808726 


1.917626 


5 


1.591072 


1.640606 


1.744146 


1.853944 


1.970361 


6 


1.620905 


1.673418 


1.783389 


1.900293 


2.024546 


7 


1.651297 


1.706886 


1.823516 


1.947800 


2.080221 


8 


1.682259 


1.741024 


1.864545 


1.996495 


2.137427 


9 


1.713800 


1.775845 


1.906497 


2.046407 


2.196206 


30 


1.745931 


1.811362 


1.949393 


2.097568 


2.256602 


1 


1.778671 


1.847589 


1.993255 


2.150007 


2.318658 


2 


1.812021 


1.884541 


2.038103 


2.203757 


2.382421 


3 


1.845996 


1.922231 


2.083960 


2.258851 


2.447937 


4 


1.880687 


1.960676 


2.130849 


2.315322 


2.515256 


5 


1.915870 


1.999889 


2.178973 


2.373205 


2.584426 


6 


1.951343 


2.039887 


2.227816 


2.432535 


2.655497 


7 


1.988388 


2.080685 


2.277942 


2.493349 


2.728524 


8 


2.025671 


2.122299 


2.329196 


2.555682 


2.803558 


9 


2.063647 


2.164745 


2.381603 


2.619574 


2.880656 


40 


2.102345 


2.208040 


2.435189 


2.685064 


2959874 


1 


2.141764 


2.252200 


2.489981 


2.752190 


3.041270 


2 


2.181922 


2.297244 


2.546005 


2.820995 


3.124905 


3 


2.222833 


2.343189 


2.603290 


2.891520 


3.210840 


4 


2.264511 


2.390053 


2.661864 


2.963808 


3.299138 


5 


2.306960 


2.437854 


2.721756 


3.037903 


3.389865 


6 


2.350226 


2.486611 


2.782996 


3.113851 


3.483086 


7 


2.394292 


2.536344 


2.845613 


3.191697 


3.578871 


8 


2.439185 


2.587070 


2.909639 


3.271490 


3.677289 


9 


2.484920 


2.638812 


2.975106 


3.353277 


3.778415 


50 


2.531512 


2.691588 


3.042046 


3.437108 


3.882321 


1 


2.578978 


2.745420 


3.110492 


3.523036 


3.989086 


2 


2.627331 


2.800328 


3.180478 


3.611112 


4.098785 


3 


2.676595 


2.856334 


3.252039 


3.701390 


4.211502 


4 


2.726781 


2.913461 


3.325210 


3.793925 


4.327318 


5 


2.777989 


2.971730 


3.400027 


3.888773 


4.446319 


6 


2.829930 


3.031165 


3.476528 


3.985992 


4.568593 


7 


2.883057 


3.091788 


3.554750 


4.085642 


4.694231 


8 


2.937114 


3.153624 


3.634732 


4.187783 


4.823320 


9 


2.992185 


3/216696 


3.716513 


4.292478 


4.955962 


60 


3.048288 


3.281030 


3.800135 


4.399790 


5.092251 



COMPOUND INTEREST. 

TABLE NO. II. Continued. 

The Amount of One Dollar at Compound Interest for the 
Time and at the Rate Per Cent Stated. (14-i) n 



67 



Periods 


3% 


3 1-2 % 


4% 


4 1-2 % 


5% 


1 


1.030000 


1.035000 


1.040000 


1.O45000 


1.050000 


2 


1.060900 


1.071225 


1.081600 


1.092025 


1.102500 


3 


1.092727 


1.108718 


1.124864 


1.141166 


1.157625 


4 


1.125509 


1.147523 


1.169859 


1.192519 


1.215506 


5 


1.159274 


1.187686 


1.216653 


1.246182 


1.276282 


6 


1.194052 


1.229255 


1.265319 


1.302260 


1.340096 


7 


1.229874 


1.272279 


1.315932 


1.360862 


1.407100 


8 


1.266770 


1.316809 


1.368569 


1.422101 


1.477455 


9 


1.304773 


1.362897 


1.423312 


1.486095 


1.551328 


10 


1.343916 


1.410599 


1.480244 


1.552969 


1.628894 


1 


1.384234 


1.459969 


1.539454 


1.622853 


1.710339 


2 


1.425761 


1.511069 


1.601032 


1.695881 


1.795856 


3 


1.468534 


1.563956 


1.665074 


1.772196 


1.885649 


4 


1.512590 


1.618695 


1.731676 


1.851945 


1.979932 


5 


1.557967 


1.675349 


1.800944 


1.935282 


2.078928 


6 


1.604706 


1.733986 


1.872981 


2.022370 


2.182875 


7 


1.652848 


1.794676 


1.947901 


2.113377 


2.292018 


8 


1.702433 


1.857489 


2.025817 


2.208479 


2.406619 


9 


1.753506 


1.922501 


2.106849 


2.307850 


2.526950 


20 


1.806111 


1.989789 


2.191123 


2.411714 


2.653298 


1 


1.860295 


2.059431 


2.278768 


2.520241 


2.785963 


2 


1.916103 


2.131512 


2.369918 


2.633652 


2.925261 


3 


1.973587 


2.206114 


2.464716 


2.752166 


3.071524 


4 


2.032794 


2.283328 


2.563304 


2.876014 


3.225100 


5 


2.093778 


2.363245 


2.665836 


3.005434 


3.386355 


6 


2.156591 


2.445959 


2.772470 


3.140679 


3.555673 


7 


2.221289 


2.531567 


2.883369 


3.282010 


3.733456 


8 


2.287928 


2.620172 


2.998703 


3.429700 


3.920129 


9 


2.356566 


2.711878 


3.118651 


3.584036 


4.116136 


30 


2.427262 


2.806794 


3.243397 


3.745318 


4.321942 


1 


2.500080 


2.905031 


3.373133 


3.913857 


4.538039 


2 


2.575083 


3.006708 


3.508059 


4.089981 


4.764942 


3 


2.652335 


3.111942 


3.648381 


4.274030 


5.003189 


4 


2.731905 


3.220860 


3.794316 


4.466362 


5.253348 


5 


2.813862 


3.333590 


3.946089 


4.667348 


5.516015 


6 


2.898278 


3.450266 


4.103933 


4.877378 


5.791816 


7 


2.985227 


3.571025 


4.268090 


5.096860 


6.081407 


8 


3.074783 


3.696011 


4.438813 


5.326219 


6.385477 


9 


3.167027 


3.825372 


4.616366 


5.565899 


6.704751 


40 


3.262038 


3.959260 


4.801021 


5.816365 


7.039989 


1 


3.359899 


4.097833 


4.993061 


6.078101 


7.391988 


2 


3.460696 


4.241258 


5.192784 


6.351615 


7.761588 


3 


3.564517 


4.389702 


5.400495 


6.637438 


8.149667 


4 


3.671452 


4.543342 


5.616515 


6.936123 


8.557150 


5 


3.781596 


4.702359 


5.841176 


7.248248 


8.985008 


6 


3.895044 


4.866941 


6.074823 


7.574420 


9.434258 


7 


4.011895 


5.037285 


6.317816 


7.915268 


9.905971 


8 


4.132252 


5.213589 


6.570528 


8.271456 


10.401270 


9 


4.256219 


5.396065 


6.833349 


8.643671 


10.921333 


50 


4.383906 


5.584927 


7.106683 


9.032636 


1 11.467400 


1 


4.515423 


5.780399 


7.390951 


9.439105 


12.040770 


2 


4.650886 


5.982713 


7.686589 


9.863865 


12.642808 


3 


4.790412 


6.192108 


7.994057 


10.307739 


13.274949 


4 


4.934125 


6.408832 


8.313814 


10.771587 


13.938696 


5 


5.082149 


6.633141 


8.646367 


11.256308 


14.635631 


6 


5.234613 


6.865301 


8.992222 


11.762842 


15.367412 


7 


5.391651 


. 7.105587 


9.351910 


12.292170 


16.135783 


8 


5.553401 


7.354282 


9.725987 


12.845318 


16.942572 


9 


5.720003 


7.611682 


10.115026 


13.423357 


17.789701 


60 


5.891603 


7.878091 


10.519627 


14.027408 


18.679186 



68 



FINANCE AND LIFE INSURANCE. 



TABLE NO. II. Concluded. 

The Amount of One Dollar at Compound Interest for the 
Time and at the Rate Per Cent Stated. (l+i)° 



Periods 


6% 


7% 


8% 


9% 


10% 


1 


1.060000 


1.070000 


1.080000 


1.090000 


1.100000 


2 


1.123600 


1.144900 


1.166400 


1.188100 


1.210000 


3 


1.191016 


1.225043 


1.259712 


1.295029 


1.441000 


4 


1.262477 


1.310796 


1.360489 


1.411582 


1.464100 


5 


1.338226 


1.402552 


1.469328 


1.538624 


1.610510 


6 


1.418519 


1.500730 


1.586874 


1.677100 


1.771561 


7 


1.503630 


1.605782 


1.713824 


1.828039 


1.948717 


8 


1.593848 


1.718186 


1.850930 


1.992563 


2.143589 


9 


1|689479 


1.838459 


1.999005 


2.171893 


2.357948 


10 


1.790848 


1.967151 


2.158925 


2.367364 


2.593743 


1 


1.898299 


2.104852 


2.331639 


2.580426 


2.853117 


2 


2.012196 


2.252192 


2.518170 


2.812665 


3.138428 


3 


2.132928 


2.409845 


2.719624 


3.065805 


3.452271 


4 


2.260904 


2.578534 


2.937194 


3.341727 


3.797498 


5 


2.396558 


2.759031 


3.172169 


3.642483 


4.177248 


6 


2.540352 


2.952164 


3.425943 


3.970306 


4.594973 


7 


2.692773 


3.158815 


3.700018 


4.327633 


5.054470 


8 


2.854339 


3.379932 


3.996019 


4.717120 


5.559917 


9 


3.025600 


3.616528 


4.315701 


5.141661 


6.115909 


20 


3.207135 


3.869684 


4.660957 


5.604411 


6.727500 


1 


3.399564 


4.140562 


5.033834 


6.108808 


7.400250 


2 


3.603537 


4.430402 


5.436540 


6.658600 


8.140275 


3 


3.819750 


4.740530 


5.871464 


7.257875 


8.954302 


4 


4.048935 


5.072367 


6.341181 


7.911083 


9.849733 


5 


4.291871 


5.427433 


6.848475 


8.623081 


10.834706 


6 


4.549383 


5.807353 


7.396353 


9.399158 


11.918177 


7 


4.822346 


6.213868 


7.988062 


10.245082 


13.109994 


8 


5.111687 


6.648838 


8.627106 


11.167139 


14.420994 


9 


5.418388 


7.114257 


9.317275 


12.172182 


15.863093 


30 


5.743491 


7.612255 


10.062657 


13.267679 


17.449402 


1 


6.088101 


8.145113 


10.867669 


14.461770 


19.194343 


2 


6.453387 


8.715271 


11.737083 


15.763329 


21.113777 


3 


6.840590 


9.325340 


12.676050 


17.182028 


23.225154 


4 


7.251025 


9.978114 


13.690134 


18.728411 


25.547670 


5 


7.686087 


10.676581 


14.785344 


20.413968 


28.102437 


6 


8.147252 


11.423942 


15.668172 


22.251225 


30.912681 


7 


8.636087 


12.223618 


17.245626 


24.253835 


34.003949 


8 


9.154252 


13.079271 


18.625276 


26.436681 


37.404343 


9 


9.703508 


13.994820 


20.115298 


28.815982 


41.144778 


40 


10.285718 


14.974458 


21.724522 


31.409420 


45.259256 


1 


10.902861 


16.022670 


23.462483 


34.236268 


49.785181 


2 


11.557033 


17.144257 


25.339482 


37.317532 


54.763699 


3 


12.250455 


18.344355 


27.366640 


40.676110 


60.240069 


4 


12.985482 


19.628459 


29.555972 


44.336960 


66.264076 


5 


13.764611 


21.002452 


31.920449 


48.327286 


72.890484 


6 


14.590487 


22.472623 


34.474085 


52.676742 


80.179532 


7 


15.465917 


24.045707 


37.232012 


57.417649 


88.197485 


8 


16.393872 


25.728907 


40.210573 


62.585237 


97.017234 


9 


17.377504 


27.529930 


43.427419 


68.217908 


106.718957 


50 


18.420154 


29.457025 


46.901612 


74.357520 


117.390853 


1 


19.525364 


31.519017 


50.653742 


81.049697 


129.129938 


2 


20.696885 


33.725348 


54.706041 


88.344170 


142.042932 


3 


21.938698 


36.086122 


59.082524 


96.295145 


156.247225 


4 


23.255020 


38.612151 


63.809126 


104.961708 


171.871948 


5 


24.650322 


41.315002 


68.913856 


114.408262 


189.059143 


6 


26.129341 


44.207052 


73.816965 


124.705006 


207.965057 


7 


27.697101 


47.300155 


79.722322 


135.928456 


228.761562 


8 


28.358927 


.50.612653 


86.100107 


148.162017 


251.637718 


9 


31.120463 


54.155539 


92.988116 


161.496599 


276.801490 


60 


32.987691 


57.946427 


100.427165 


176.031292 


304.481639 



70 



finance Sand life insurance. 



TABLE NO. III. 

The Present Value of One Dollar Payable at Any One of 

the Given Periods, Discounted at the Rate Per Cent 

Stated. v u 



Periods 


1-4% 


1-3% 


3-8% 


5-12 % 


1-2 % " 





1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


1 


.997506 


.996678 


.996264 


.995851 


.995025 


2 


.995019 


.993366 


.992619 


.991718 


.990075 


3 


.992537 


.990066 


.988834 


.987603 


.985149 


4 


.990062 


.986777 


.985140 


.983505 


.980248 


5 


.987593 


.983499 


.981457 


.979425 


.975370 


6 


.985130 


.980231 


.977793 


.975361 


.970518 


7 


.982674 


.976975 


.974140 


.971313 


.965690 


8 


.980223 


.973729 


.970434 


.967283 


.960885 


9 


.977778 


.970494 


.966875 


.963265 


.956105 


10 


.975340 


.967270 


.963263 


.959272 


.951348 


1 


.972906 


.964056 


.959662 


.955292 


.946615 


2 


.970781 


.960853 


.956080 


.951328 


.941905 


3 


.968061 


.957661 


.952508 


.947381 


.937219 


4 


.965647 


.954479 


.948949 


.943449 


.932556 


5 


.963239 


.951308 


.945402 


.939535 


.927917 


6 


.960837 


.948148 


.941872 


.935636 


.923300 


7 


.958441 


.944998 


.938354 


.931754 


.918707 


8 


.956050 


.941858 


.934848 


.927888 


.914136 


9 


.953666 


.938729 


.931555 


.924038 


.909588 


20 


.951288 


.935611 


.927876 


.920204 


.905063 


1 


.948916 


.932502 


.924409 


.916386 


. 900560 


2 


.946549 


.929404 


.920956 


.912583 


.896080 


3 


.944189 


.926317 


.917516 


.908793 


.891622 


4 


.941834 


.923239 


.914088 


.905025 


.887186 


5 


.339485 


.920172 


.910673 


.901270 


.882772 


6 


.937142 


.917115 


.907271 


.897530 


.878380 


7 


.934808 


.914068 


.903881 


.893806 


.874010 


8 


.932474 


.911031 


.900505 


.890098 


.869662 


9 


.930147 


.908005 


.897141 


.886404 


.865335 


10 


.927829 


.904988 


.893789 


.882726 


.861030 


1 


.925515 


.901987 


.890450 


.879063 


.856746 


2 


.923217 


.898985 


.887123 


.875416 


.852484 


3 


.920903 


.895998 


.887809 


.871783 


.848242 


4 


.918608 


.893021 


.880507 


.868166 


.844022 


5 


.916318 


.890054 


.877218 


.864564 


.839823 


6 


.914032 


.887097 


.873948 


.860976 


.835645 


7 


.911753 


.884150 


.870676 


.857404 


.831487 


8 


.909479 


.881213 


.867423 


.853846 


.827351 


9 


.907211 


.878285 


.864183 


.850303 


.823235 


40 


.904939 


.875367 


.860954 


.846775 


.819139 


1 


.902692 


.872459 


.857738 


.843261 


.815064 


2 


.900441 


.869561 


.854533 


.839762 


.811009 


3 


.898195 


.866672 


.851341 


.836278 


.806974 


4 


.895955 


.863792 


.848161 


.832808 


.802959 


5 


.893721 


.860923 


.844992 


.829352 


.798964 


6 


.891492 


.858062 


.841835 


.825911 


.794989 


7 


.889269 


.855212 


.838690 


.822484 


.791034 


8 


.887051 


.852371 


.835556 


.819071 


.787098 


9 


.884839 


.849539 


.832435 


.815672 


.783183 


50 


.882634 


.846716 


.829326 


.812287 


.779286 


1 


.880432 


.843903 


.826227 


.808917 


.775409 


2 


.878236 


.841099 


.823131 


.805561 


.771551 


3 


.876046 


.838305 


.820066 


.802218 


.767713 


4 


.873861 


.835520 


.817002 


.798890 


.763893 


5 


.871682 


.832744 


.813950 


.795574 


.760093 


6 


.869508 


.829978 


.810909 


.792273 


.756312 


7 


.867340 


.827220 


.807878 


.788986 


.752548 


8 


.865177 


.824472 


.804861 


.785712 


.748804 


9 


.863019 


.821733 


.801855 


.782452 


.745079 


60 


.860868 


.819003 


.798852 


.779205 


.741372 



COMPOUND DISCOUNT. 



71 



TABLE NO. III. Continued. 

The Present Value of One Dollar Payable at Any One of 

the Given Periods, Discounted at the Rate Per Cent 

Stated. v a 



Periods 


1-4% 


1-5% 


3-8% 


5-12 % 


1-2% 


61 


.858722 


.816284 


.795865 


.775972 


.737684 


2 


.856581 


.813570 


.792895 


.772752 


.734014 


3 


.854444 


.810867 


.789932 


.769546 


.730362 


4 


.852314 


.808173 


.786981 


.766353 


.726728 


5 


.850188 


.805488 


.784041 


.763173 


.723113 


6 


.848068 


.802812 


.781116 


.760006 


.719515 


7 


.845953 


.800145 


.778194 


.756853 


.715935 


8 


.843844 


.797487 


.775286 


.753712 


.712374 


9 


.841739 


.794838 


.772390 


.750585 


.708829 


70 


.839640 


.792194 


.769504 


.747470 


.705303 


1 


.837546 


.789565 


.766629 


.744369 


.701794 


2 


.835458 


.786942 


.763765 


.741280 


.698302 


3 


.833375 


.784327 


.760912 


.738204 


.694828 


4 


.831296 


.781722 


.758069 


.735141 


.691371 


5 


.829223 


.779125 


.755237 


.732091 


.687932 


6 


.827155 


.776536 


.752415 


.729053 


.684509 


7 


.825093 


.773956 


.749604 


.726028 


.681104 


8 


.823035 


.771388 


.746804 


.723015 


.677715 


9 


.820982 


.768822 


.744014 


.720015 


.674343 


80 


.818935 


.766268 


.741234 


.717024 


.670988 


1 


.816893 


.763722 


.738465 


.714053 


.667650 


2 


.814846 


.761185 


.735706 


.711090 


.664329 


3 


.812824 


.758656 


.732957 


.708139 


.661023 


4 


.810794 


.756136 


.730319 


.705201 


.657735 


5 


.808775 


.753624 


.727491 


.702275 


.654462 


6 


.806015 


.751120 


.724773 


.699361 


.651206 


7 


.804746 


.748625 


.722065 


.696459 


.647967 


8 


.802737 


.746137 


.719368 


.693569 


.644743 


9 


.800737 


.743659 


.716680 


.690791 


.641535 


90 


.798741 


.741188 


.714003 


.687825 


.638343 


1 


.796749 


.738725 


.711335 


.684971 


.635168 


2 


.794762 


.736271 


.708677 


.682129 


.632008 


3 


.792779 


.733825 


.706030 


.679298 


.628863 


4 


.790803 


.731387 


.703392 


.676480 


.625735 


5 


.788841 


.728957 


.700764 


.673673 


.622621 


6 


.786863 


.726536 


.698146 


.670877 


.619524 


7 


.784901 


.724122 


.695538 


.668094 


.616442 


8 


.782944 


.721716 


.692939 


.665321 


.613375 


9 


.780991 


.719320 


.690351 


.662561 


.610323 


100 


.779044 


.716929 


.687772 


.659476 


.607287 


1 


.777101 


.714547 


.685202 


.657074 


.604265 


2 


.775163 


.712173 


.682642 


.654347 


.601259 


3 


.773230 


.709807 


.680092 


.651632 


.598268 


4 


.771302 


.707450 


.677551 


.648928 


.595291 


5 


.769378 


.705098 


.675019 


.646236 


.592330 


6 


.767460 


.702756 


.672498 


.643554 


.589383 


7 


.765546 


.700421 


.669986 


.640884 


.586451 


8 


.763637 


.698094 


.667482 


.638225 


.583533 


9 


.761732 


.695775 


.664989 


.635576 


.580630 


110 


.759833 


.693463 


.662504 


.632939 


.577741 


1 


.757937 


.691159 


.660029 


.630313 


.574867 


2 


.756048 


.688863 


.657563 


.627698 


.572007 


3 


.754162 


.686575 


.655107 


.625093 


.569161 


4 


.752281 


.684294 


.652659 


.622499 


.566329 


5 


.750406 


.682020 


.650221 


.619916 


.563512 


6 


.748534 


.679754 


.647792 


.617344 


.560708 


7 


.746668 


.677496 


.645371 


.614782 


.557919 


8 


.744806 


.675245 


.642960 


.612231 


.555143 


9 


.742948 


.673002 


.640602 


.609691 


.552381 


120 


.741096 


.670768 


.638165 


.607161 


.549633 



72 



FINANCE AND LIFE INSURANCE. 



TABLE NO. III. Continued. 

The Present Value of One Dollar Payable at Any One ot 

the Given Periods, Discounted at the Rate Per Cent 

Stated. v n 



Periods 



3-4 



7-8% 



1% 



1 1-8 % 






1.000000 


1.000000 


1.000000 


1.000000 | 


1.000000 


1 


.993789 


.992556 


.991326 


.990099 


.988875 


2 


.987616 


.985167 


.982727 


.980296 


.977874 


3 


.981482 


.977834 


.974203 


.970590 


.966995 


4 


.975387 


.970551 


.965752 


.960980 


.956238 


5 


.969327 


.963330 


.957375 


.951466 


.945600 


6 


.963307 


.956159 


.949071 


.942045 


.935080 


7 


.957324 


.949041 


.940839 


.932718 


.924678 


8 


.951377 


.941976 


.932678 


.923483 


.914391 


9 


.945468 


.934964 


.924588 


.914340 


.904219 


10 


.939596 


.928004 


.916568 


.905287 


.894159 


1 


.733759 


.921096 


.908617 


.896324 


.884214 


2 


.927960 


.914239 


.900736 


.887449 


.874376 


3 


.922196 


.907434 


.892923 


.876663 


.864648 


4 


.916468 


.900679 


.885177 


.869963 


.855029 


5 


.910860 


.893974 


.877499 


.816349 


.845517 


6 


.905119 


.887319 


.869888 


.852821 


.836111 


7 


.899497 


.880714 


.862342 


.844377 


.826809 


8 


.893910 


.874158 


.854862 


.836017 


.817611 


9 


.888359 


.867651 


.847447 


.827740 


.808516 


20 


.882840 


.861190 


.840096 


.819544 


.799466 


1 


.877356 


.854781 


.832809 


.811430 


.790627 


2 


.871907 


.848418 


.825585 


.803396 


.781831 


3 


.866491 


.842102 


.818424 


.795442 


.773134 


4 


.861110 


.835834 


.811325 


.787566 


.764523 


5 


.855770 


.829612 


.804287 


.779768 


.756027 


6 


.850446 


.823436 


.797311 


.772048 


.747617 


7 


.845164 


.817307 


.790395 


.764404 


.739299 


8 


.839914 


.811223 


.783539 


.756836 


.731076 


9 


.834697 


.805184 


.776743 


.749342 


.722942 


30 


.829512 


.799190 


.770005 


.741923 


.714899 


1 


.824360 


.793241 


.763326 


.734577 


.706946 


2 


.819240 


.787336 


.756705 


.727304 


.699083 


3 


.814152 


.781475 


.750141 


.720103 


.691305 


4 


.809095 


.775657 


.743634 


.712973 


.683624 


5 


.804069 


.769884 


.737184 


.705914 


.676009 


6 


.799075 


.764152 


.730789 


.698925 


.668488 


7 


.794112 


.758462 


.724451 


.692005 


.661052 


8 


.789179 


.752818 


.718167 


.685153 


.653699 


9 


.784278 


.747212 


.711937 


.678369 


.646425 


40 


.779406 


.741652 


.705762 


.671653 


.639234 


1 


.774564 


.736131 


.699640 


.665003 


.632123 


2 


.769755 


.730651 


.693571 


.658419 


.625091 


3 


.764973 


.725212 


.687555 


.651900 


.618137 


4 


.760222 


.719814 


.681591 


.645445 


.611260 


5 


.755500 


.714455 


.675679 


.639055 


.604460 


6 


.750807 


.709137 


.669818 


.632728 


.597735 


7 


.746144 


.703858 


.664008 


.626463 


.591086 


8 


.741510 


.698618 


.658248 


.620260 


.584541 


9 


.736904 


.693418 


.652539 


.614119 


.578007 


50 


.732327 


.688256 


.646878 


.608039 


.571577 


1 


.727778 


.683133 


.642170 


.602019 


.565219 


2 


.723258 


.678047 


.635707 


.596058 


.558931 


3 


.718766 


.672999 


.630193 


.590157 


.552713 


4 


.714301 


.667990 


.624725 


.584314 


.546564 


5 


.709874 


.663018 


.619306 


.5*78528 


.540484 


. 6 


.705455 


.658082 


.613923 


.572800 


.534471 


7 


.701074 


.653183 


.608607 


.567129 


.528525 


8 


.696719 


.'648321 


.603328 


.561514 


'.522645 


9 


.692392 


.643495 


.598094 


.555955 


.516831 


60 


.688191 


.638705 


.592904 


.550450 


.511081 



COMPOUND DISCOUNT. 



73 



TABLE NO. III. Continued. 

The Present Value of One Dollar Payable at Any One of 

the Given Periods, Discounted at the Rate Per Cent 

Stated. v n 



Periods 


1 1-4 % 


1 3-8 % 


1 1-2 % 


1 5-8 % 


1 3-4 % 





1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


1 


.987654 


.986437 


.985222 


.984010 


.982801 


2 


.975463 


.973057 


.970662 


.968274 


.965898 


3 


.963419 


.959859 


.956317 


.952792 


.949285 


4 


.951525 


.946840 


.942184 


.937557 


.932959 


5 


.939777 


.933998 


.928260 


.922565 


.916913 


6 


.928775 


.921330 


.914542 


.907813 


.901143 


7 


.916716 


.908833 


.901027 


.893297 


.885644 


8 


.905499 


.896501 


.887711 


.879013 


.870412 


9 


.894221 


.884347 


.874592 


.764957 


.855441 


10 


.883182 


.872352 


.861667 


.851126 


.840729 


1 


.872278 


.860520 


.848933 


.837517 


.816269 


2 


.861509 


.848849 


.836387 


.824124 


.812058 


3 


.850873 


.837336 


.824027 


.810946 


.798091 


4 


.840369 


.825979 


.811849 


.797979 


.784365 


5 


.829994 


.814776 


.799852 


.785219 


.770875 


6 


.819747 


.803726 


.788031 


.772664 


.757616 


7 


.809627 


.792823 


.776385 


.760308 


.744586 


8 


.799632 


.782070 


.764912 


.748151 


.731779 


9 


.789760 


.771463 


.753607 


.731289 


.719194 


20 


.780100 


.760999 


.742470 


.724416 


.706825 


1 


.770380 


.750677 


.731498 


.712932 


.694668 


2 


.760869 


.740495 


.720688 


.701434 


.682720 


3 


.751476 


.730452 


.710037 


.690216 


.670978 


4 


.742198 


.720545 


.699544 


.679181 


.659438 


5 


.733034 


.710771 


.689206 


.668321 


.648096 


6 


.723986 


.701131 


.679021 


.657634 


.636950 


7 


.715048 


.691621 


.668986 


.647118 


.625995 


8 


.706220 


.681241 


.659099 


.636771 


.615228 


9 


.697501 


.672987 


.649359 


.626589 


.604647 


30 


.688890 


.663759 


.639762 


.616569 


.694248 


1 


.680386 


.654855 


.630308 


.606710 


.584027 


2 


.671986 


.645973 


.620993 


.597009 


.573982 


3 


.663689 


.637212 


.611816 


.587463 


.564111 


4 


.655496 


.628569 


.602774 


.578069 


.554408 


5 


.647403 


.620043 


.593866 


.568825 


.544873 


6 


.639411 


.611633 


.585090 


.559730 


.535502 


7 


.631511 


.603338 


.576443 


.550780 


.526292 


8 


.623720 


.595154 


.567924 


.541972 


.517240 


9 


.616020 


. 587082 


.559531 


.533306 


.508344 


40 


.608415 


.579114 


.551262 


.524778 


.499601 


1 


.600904 


.571265 


.543116 


.516387 


.491008 


2 


.593472 


.563516 


.535089 


.508130 


.482563 


3 


.586158 


.555873 


.527182 


.501158 


.474264 


4 


.578922 


.548333 


.519391 


.492009 


.466107 


5 


.571775 


.540896 


.511715 


.484142 


.458090 


6 


.564716 


.533560 


.504153 


.476401 


.450212 


7 


.557744 


.526323 


.496702 


.468883 


.442469 


8 


.550859 


.519184 


.489362 


.461287 


.434858 


9 


.544058 


.512143 


.482130 


.453911 


.427379 


50 


.537341 


.505196 


.475005 


.446654 


.420029 


1 


.530707 


.498344 


.467982 


.439511 


.412805 


2 


.524155 


.491585 


.461066 


.432483 


.405705 


3 


.517684 


.484917 


.454252 


.425567 


.398724 


4 


.511293 


.478340 


.447538 


.418763 


.391869 


5 


.504980 


.471852 


.440925 


.412066 


.385129 


6 


.498748 


.465452 


.434419 


.405476 


.379505 


7 


.492589 


.459139 


.427989 


.398993 


.371995 


8 


.486508 


.453016 


.421664 


.392614 


.365597 


9 


.480502 


.446769 


.415430 


.386356 


.359309 


60 


.474569 


.440708 


.409293 


.380159 


.353130 



74 



FINANCE AND LIFE INSURANCE. 



TABLE NO. III. Continued. 

The Present Value of One Dollar Payable at Any One of 

the Given Periods, Discounted at the Rate Per Cent 

Stated. v n 



Periods 


1 7-8 % 


2% 


2 1-4 % 


2 1-2 % 


2 3-4 % 


o 1 


1.000000 | 


1.000000 


1.000000 | 


1.000000 


1.000000 


1 


.981595 


.980392 


.977995 


.975609 


.973236 


2 1 


.963529 


.961169 


.956474 


.951814 


.947188 


3 


.945795 


.942322 


.935427 


.928599 


.921838 


4 


.928387 


.923845 


.914843 


.905951 


.897166 


5 


.911300 


.905731 


.894712 


.883854 


.873154 


6 


.894528 


.887971 


.875024 


.862297 


.849785 


7 


.878064 


.870560 


.855769 


.841265 


.827041 


8 


.861903 


.853490 


.836938 


.820747 


.804906 


9 


.846040 


.836755 


.818522 


.800728 


.783364 


10 


.830468 


.820348 


.800510 


.781198 


.726398 


1 


.815183 


.804263 


.782895 


.762145 


.741993 


2 


.800180 


.788493 


.765667 


.743556 


.722134 


3 


.785453 


.773033 


.748819 


.725420 


.702807 


4 


.770998 


.757875 


.732341 


.707727 


.683997 


5 


.756806 


.743015 


.716226 


.690466 


.665691 


6 


.742877 


.728446 


.700466 


.673625 


.647874 


7 


.729204 


.714163 


.685052 


.657195 


.630535 


8 


.715783 


.700159 


.669978 


.641166 


.613658 


9 


.702609 


.686431 


.655235 


.625528 


.597235 


20 


.689678 


.672971 


.640816 


.610271 


.681251 


1 


.676984 


.659776 


.626715 


.595386 


.565694 


2 


.664514 


.646839 


.612952 


.580865 


.550554 


3 


.652293 


.634156 


.599437 


.566697 


.535819 


4 


.640288 


.621721 


.586247 


.552875 


.521478 


5 


.628503 


.609531 


.573346 


.539391 


.507521 


6 


.616936 


.597597 


.560730 


.526235 


.493937 


7 


.605581 


.585862 


.548391 


.513400 


.480718 


8 


.594425 


.574374 


.536324 


.500878 


.467852 


9 


.583494 


.563112 


.524522 


.488661 


.455331 


30 


.572756 


.552071 


.512980 


.476743 


.443144 


1 


.562215 


.541246 


.501692 


.465115 


.431284 


2 


.551866 


.530633 


.490652 


.453771 


.419741 


3 


.541708 


.520229 


.479856 


.442703 


.408507 


4 


.531739 


.510028 


.469296 


.431905 


.397573 


5 


.521952 


.500028 


.458969 


.421371 


.386933 


6 


.512345 


.490223 


.448870 


.411094 


.376577 


7 


.502916 


.480611 


.438993 


.401067 


.366499 


8 


.493656 


.471187 


.429333 


.391285 


.356690 


9 


.484574 


.461948 


.419885 


.381741 


.347143 


40 


.475655 


.452890 


.410646 


.372431 


.337852 


1 


.466900 


.444010 


.401610 


.363347 


.328810 


2 


.458307 


.435304 


.392772 


.354485 


.320010 


3 


.449872 


.426769 


.384129 


.345839 


.311445 


4 


.441592 


.418401 


.375676 


.337404 


.303109 


5 


.433465 


.410197 


.367410 


.329174 


.294997 


6 


.425487 


.402154 


.359325 


.321146 


.287102 


7 


.417655 


.394268 


.351418 


.313313 


.279418 


8 


.409969 


.386538 


.343685 


.305671 


.271939 


9 


.402423 


.378958 


.336122 


.298216 


.264661 


50 


.395016 


.371528 


.328726 


.290942 


.257578 


1 


.387746 


.364246 


.321492 


.283845 


.250795 


2 


.380610 


.357104 


.314418 


.276922 


.243986 


3 


.373604 


.350102 


.307499 


.270168 


.237456 


4 


.366728 


.343237 


.300733 


.263578 


.231101 


5 


.359978 


.336507 


.294115 


.257150 


.224905 


6 


.353353 


.329909 


.287643 


.250877 


.218896 


7 


.346849 


.323440 


.281313 


.244759 


.213027 


8 


.340466 


.317098 


.275124 


.238789 


.207326 


9 


.334199 


.310881 


.269070 


.232964 


.201777 


60 


.328048 


.304782 


| .263149 


.227284 


| .196377 



COMPOUND DISCOUNT. 



75 



TABLE NO. ill. Continued. 

The Present Value of One Dollar Payable at Any One of 
the Given Periods, Discounted at the Rate Per Cent 
Stated. v a 



Periods 


3% 


3 1-2 % 


4% 


4 1-2 % 


5% 


J 


1.000000 


1.000000 | 


1.000000 i 


1.000000 


1.000000 


1 


.970874 


.966184 | 


.961538 | 


.956938 


.952381 


2 


.942596 


.933511 j 


.924556 


.915730 


.907029 


3 


.915142 


.901943 


.888996 


.876297 


.863838 


4 


.888487 


.871442 ] 


.854804 


.838561 


.822702 


5 


.862609 


.841973 | 


.821927 


.802451 


.783526 


6 


.837484 


.813501 | 


.790315 


.767896 


.746215 


7 


.813092 


.785991 


.759918 


.734828 


.710681 


8 


.789409 


.759412 


.730690 


.703185 


.676839 


9 


.766417 


.733731 


.702587 


.672904 


.644609 


10 


.744094 


.708919 


.675564 


.643928 


.613913 


1 


.722421 


.684946 


.649581 


.616199 


.584679 


2 


.701380 


.661783 


.624597 


.589664 


.556837 


3 


.680951 


.639404 


.600574 


.564272 


.530321 


4 


.661118 


.617782 


.577475 


.539973 


.505068 


5 


.641862 


.596891 


.555265 


.516720 


.481017 


6 


.623167 


.576706 


.533908 


.494469 


.458112 


7 


.605016 


.557204 


.513373 


.473176 


.436297 


8 


.587395 


.538361 


.493628 


.452800 


.415521 


9 


.570286 


.520156 


.474642 


.433302 


.395734 


20 


.553676 


.502566 


.456387 


.414643 


.376889 


1 


.537549 


.485571 


.438834 


.396787 


.358942 


2 


.521893 


.469151 


.421955 


.379701 


.341850 


3 


.506692 


.452286 


.405726 


.363350 


.325571 


4 


.491934 


.437957 


.390121 


.347703 


.310068 


5 


.477606 


.423147 


.375117 


.332731 


.295303 


6 


.463695 


.408838 


.360689 


.318402 


.281241 


7 


.450189 


.395012 


.346817 


.304691 


.267848 


8 


.437077 


.381654 


.334477 


.291571 


.255094 


9 


.424346 


.368748 


.320651 


.279015 


.242946 


30 


.411987 


.356278 


.308319 


.267000 


.231377 


1 


.399987 


.344230 


.296460 


.255502 


.220359 


2 


.388337 


.332590 


.285058 


.244500 


.209866 


3 


.377026 


.321343 


.274094 


.233971 


.199873 


4 


.366045 


.310476 


.263552 


.223896 


.190355 


5 


.355383 


.299977 


.253415 


.214254 


.181290 


6 


.345032 


.289833 


.243669 


.205028 


.172657 


7 


.334983 


.280032 


.234297 


.196199 


.164436 


8 


.325226 


.270562 


.225285 


.187750 


.156605 


9 


.315754 


.261413 


.216621 


.179665 


.149148 


40 


.306557 


.252572 


.208289 


.171929 


.142046 


1 


.297628 


.244031 


.200278 


.164525 


.135282 


2 


.288959 


.235779 


.192575 


.157440 


.128840 


3 


.280543 


.227806 


.185168 


.150661 


.122704 


4 


.272372 


.220102 


.178046 


.144173 


.116861 


5 


.264439 


.212659 


.171198 


.137964 


.111297 


6 


.256737 


.205468 


.164614 


.132023 


.105997 


7 


.249259 


.198520 


.158283 


.126338 


.100949 


8 


.241999 


.191806 


.152195 


.120898 


.096142 


9 


.234959 


.185320 


.146341 


.115692 


.091564 


50 


.228107 


.179053 


.140713 


.110710 


.087204 


1 


.221463 


.172998 


.135301 


.105942 


.083051 


2 


.215013 


.167148 


.130097 


.101380 


.079096 


3 


.208750 


.161496 


.125093 


.097014 


.075330 


4 


.202670 


.156035 


.120282 


.092837 


.071743 


5 


.196767 


.150758 


.115656 


.088839 


.068326 


6 


.191036 


.145660 


.111207 


.085013 


.065073 


7 


.185472 


.140734 


.106930 


.081353 


.061974 


8 


.180070 


.135975 


.102817 


.077849 


.059023 


9 


.174825 


.131377 


.098863 


.074497 


.056212 


60 


.169733 


.126934 


.095060 


.071289 


.053536 



76 



FINANCE AND LIFE INSURANCE. 



TABLE NO. III. Concluded. 

The Present Value of One Dollar Payable at Any One of 

the Given Periods, Discounted at the Rate Per Cent 

Stated. v n 



Periods 


6% 


7% 


8% 


9% 


10% 





1.000000 J 1.000000 


1.000000 


1.000000 


1.000000 


1 


.943396 


.934579 


.925926 


.917431 


.909091 


2 


.889996 


.873439 


.857339 


.841680 


.826446 


3 


.839619 


.816298 


.794832 


.772183 


.751315 


4 


.792094 


.762895 


.735030 


.708425 


.683031 


5 


.747258 


.712986 


.680583 


.649931 


.620922 


6 


.704961 


.666342 


.630170 


.596267 


.564584 


7 


.665057 


.622750 


.583490 


.547034 


.513258 


8 


.627412 


.582009 


.540269 


.501866 


.466507 


9 


.591898 


.543934 


.500249 


.460428 


.424098 


10 


.558395 


.508349 


.473193 


.422411 


.385538 


1 


.526788 


.475093 


.428883 


.387533 


.351494 


2 


.496969 


.444012 


.397114 


.355535 


.318631 


3 


.468839 


.414964 


.367688 


.326179 


.289664 


4 


.442301 


.387817 


.340461 


.299246 


.263331 


5 


.417265 


.362446 


.315242 


.274538 


.239393 


6 


.393646 


.338734 


.291890 


.251870 


.217629 


7 


.371364 


.316574 


.270269 


.231073 


.197844 


8 


.350344 


.295864 


.250249 


.211994 


.179859 


9 


.330513 


.276508 


.231712 


.194489 


.163508 


20 


.311805 


.258419 


.215658 


.178431 


.148644 


1 


.294155 


.241513 


.208656 


. 163698 


.135130 


2 


.277505 


.225713 


.183941 


.150182 


.122846 


3 


.261797 


.210947 


.170315 


.137781 


.111678 


4 


.246979 


.197147 


.157699 


.126405 


.101526 


5 


.232999 


.184294 


.146018 


.115968 


.092296 


6 


.219810 


.172195 


.135202 


.106393 


.083905 


7 


.207368 


.160930 


.125187 


.097608 


.076278 


8 


.195630 


.150402 


.115913 


.087584 


.069334 


9 


.184557 


.140563 


.107328 


.082155 


.063139 


30 


.174110 


.131367 


.099377 


.075371 


.057308 


1 


.164255 


.122773 


.092016 


.069146 


.052099 


2 


.154957 


.114741 


.085200 


.063437 


.047363 


3 


.146186 


.107234 


.078889 


.058199 


.043056 


4 


.137912 


.100219 


.073046 


.053394 


.039143 


5 


.130105 


.093663 


.067634 


.048985 


.035584 


6 


.122741 


.087535 


.062625 


.044941 


.032349 


7 


.115793 


.081809 


.057986 


.041229 


.029409 


8 


.109239 


.076457 


.053790 


.037826 


.026734 


9 


.103056 


.071455 


.049723 


.034702 


.024305 


40 


.097222 


.066780 


.046031 


.031837 


.022095 


1 


.091719 


.062412 


.042622 


.029208 


.020086 


2 


.086527 


.058329 


.039464 


.026797 


.018260 


3 


.081630 


.054513 


.036551 


.024584 


.016601 


4 


.077009 


.059946 


.033834 


.022554 


.015091 


5 


.072650 


.047613 


.031328 


.020692 


.013719 


6 


.068538 


.044499 


.029007 


.018983 


.012472 


7 


.064658 


.041587 


.026858 


.017416 


.011338 


8 


.060998 


.038867 


.024869 


.015978 


.010308 


9 


.057546 


.036324 


.023027 


.014659 


.009370 


50 


.054288 


.033948 


.021322 


.013448 


.008618 


1 


.051215 


.031727 


.019742 


.012338 


.007745 


2 


.048316 


.029651 


.018279 


.011319 


.007040 


3 


.045582 


.027711 


.016926 


.010385 


.006400 


4 


.043001 


.025899 


.015771 


.009527 


.005818 


5 


.040567 


.024204 


.014511 


.008741 


.005290 


6 


.038271 


.022621 


.013436 


.008019 


.004809 


7 


.036105 


.021141 


.012441 


.007357 


.004362 


8 


.034061 


.019758 


.011519 


.006749 


.003966 


9 


.032133 


.0'18465 


.011519 


.006192 


.003605 


60 


.030314 


.017257 


.009875 


.005681 


.003277 



78 



FINANCE AND LIFE INSURANCE. 



TABLE NO. IV. 

The Amount of One Dollar Payable at the End of Each 
Period, Improved at Compound Interest at the Rate 

and for the Terms Stated. Snl 



Periods 


1-4% 


1-3 % 


3-8% 


5-12 % 


1-2% 


1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


2.002500 


2.003333 


2.003750 


2.004167 


2.005000 


3 


3.007506 


3.010011 


3.011264 


3.012517 


3.015025 


4 


4.015025 


4.020044 


4.022556 


4.025069 


4.030100 


5 


5.025063 


5.033445 


5.037638 


5.041840 


5.050251 


6 


6.037626 


6.050323 


6.056529 


6.062848 


6.075502 


7 


7.052720 


7.070490 


7.079241 


7.088110 


7.105879 


8 


8.070352 


8.094058 


8.105788 


8.117646 


8.141409 


9 


9.090528 


9.121038 


9.136184 


9.151470 


9.1'2116 


10 


10.113254 


10.151441 


10.170444 


10.189601 


10.228026 


1 


11.138537 


11.185279 


11.208580 


11.232079 


11.279166 


2 


12.166383 


12.222563 


12.250612 


12.278879 


12.335562 


3 


13.196799 


13.263305 


13.296551 


13.330041 


13.397240 


4 


14.229791 


14.307516 


14.346413 


14.385584 


14.464226 


5 


15.265365 


15.355207 


15.400212 


15.445524 


15.536548 


6 


16.303528 


16.406391 


16.457960 


16.509880 


16.614230 


7 


17.344285 


17.461079 


17.519677 


17.578669 


17.697301 


8 


18.387646 


18.519284 


18.585375 


18.651913 


18.785787 


9 


19.433615 


19.581015 


19.655070 


19.729629 


19.879717 


20 


20.482199 


20.646283 


20.728776 


20.811896 


20.979115 


1 


21.533405 


21.715104 


21.806509 


21.898612 


22.084011 


2 


22.587239 


22.787487 


22.888283 


22.989856 


23.194431 


3 


23.643707 


23.863445 


23.974114 


24.085446 


24.310403 


4 


24.702816 


24.942989 


25.064018 


25.185802 


25.431955 


5 


25.764573 


26.026132 


26.158008 


26.290743 


26.559115 


6 


26.828984 


27.112886 


27.256100 


27.400288 


27.691911 


7 


27.896056 


28.203265 


28.358340 


28.514457 


28.830370 


8 


28.965797 


29.297276 


29.464683 


29.633263 


29.974522 


9 


30.038211 


30.394933 


30.575175 


30.756737 


31.124395 


30 


31.113306 


31.496249 


31.689831 


31.884890 


32.280066 


1 


32.191090 


32.601236 


32.808667 


33.017631 


33.441417 


2 


33.271568 


33.709906 


33.931699 


34.155205 


34.608624 


3 


34.354746 


34.822272 


35.058942 


35.297519 


35.781667 


4 


35.440633 


35.938346 


36.190437 


36.444593 


36.960575 


5 


36.529235 


37.058140 


37.326152 


37.596451 


38.145378 


6 


37.620558 


38.181667 


38.466144 


38.753104 


39.336105 


7 


38.714609 


39.308939 


39.610392 


39.914576 


40.532785 


8 


39.811396 


40.439968 


40.758930 


41.080888 


41.735449 


9 


40.910925 


41.574768 


41.911776 


42.252059 


42.944127 


40 


42.013202 


42.713350 


43.068948 


43.428110 


44.158847 


1 


43.118235 


43.855727 


44.230452 


44.609061 


45.379642 


2 


44.226031 


45.001912 


45.396316 


45.794933 


46.606540 


3 


45.336601 


46.151918 


46.566552 


46.985746 


47.839572 


4 


46.449943 


47.305757 


47.741176 


48.181521 


49.078770 


5 


47.567568 


48.463443 


48.920205 


49.382278 


50.324164 


6 


48.686486 


49.624987 


50.103655 


50.583038 


51.575785 


7 


49.808199 


50.790403 


51.291543 


51.798822 


52.833664 


8 


50.932716 


51.959704 


52.483886 


53.014652 


54.097832 


9 


52.060044 


53.132902 


53.680700 


54.235547 


55.368321 


50 


53.190190 


54.310011 


54.882002 


55.461529 


56.645163 


1 


54.323162 


55.491047 


56.087809 


56.692619 


57.928389 


2 


55.459066 


56.676016 


57.298138 


57.928839 


59.218028 


3 


56.597707 


57.864936 


58.513006 


59.170210 


60.514118 


4 


57.739197 


59.057819 


59.732429 


60.416754 


61.816689 


5 


58.883541 


60.254678 


60.956425 


61.668491 


63.125773 


6 


60.030746 


61.455527 


62.185011 


62.925444 


64.441402 


7 


61.180819 


62.660378 


63.418205 


64.187634 


65.763609 


8 


62.333767 


63.869246 


64.656023 


65.455084" 


67.092427 


9 


63.489597 


65.082140 


65.898483 


66.727814 


68.427889 


60 


64.648317 


66.299080 


67.145502 


68.005847 


69.770028 



ACCUMULATED ANNUITIES. 



70 



TABLE NO. IV. Continued. 

The Amount of One Dollar Payable at the End of Each 
Period, Improved at Compound Interest at the Rate 

and for the Terms Stated. Sn| 



Periods 


1-4% 


1-3 % 


3-8% 


5-12 % 


1-2% 


61 


65.809934 


67.520076 


68.407298 


69.294553 


71.118878 


2 


66.974455 


68.745143 


69.663759 


70.588628 


72.474472 


3 


68.141887 


69.974293 


70.924961 


71.888095 


73.836844 


4 


69.312238 


71.207540 


72.190863 


73.192815 


75.206028 


5 


70.485514 


72.444898 


73.461541 


74.503134 


76.582058 


6 


71.661724 


73.686381 


74.736984 


75.818906 


77.964968 


7 


72.840874 


74.932002 


76.017211 


77.140136 


79.354793 


8 


74.022972 


76.181975 


77.302238 


78.466902 


80.751565 


9 


75.208030 


77.435917 


78.592054 


79.799196 


82.155323 


70 


76.396046 


78.694036 


79.886737 


81.137042 


83.566099 


1 


77.587032 


79.956348 


81.186275 


82.480462 


84.983929 


2 


78.781023 


81.222868 


82.490687 


83.829480 


86.408849 


3 


79.977971 


82.493610 


83.799990 


85.184120 


87.840893 


4 


81.177912 


83.768588 


85.114203 


86.544403 


89.280097 


5 


82.380853 


85.047816 


86.433344 


87.910254 


90.726498 


6 


83.586801 


86.331307 


87.757432 


89.281992 


92.180130 


7 


84.795764 


87.619077 


89.086482 


90.659349 


93.641031 


8 


86.007749 


88.911140 


90.420520 


92.042445 


95.109237 


9 


87.222764 


90.207509 


91.759560 


93.431304 


96.584783 


80 


88.440817 


91.508200 


93.103590 


94.825949 


98.067707 


1 


89.661915 


92.813226 


94.452692 


96.226406 


99.558046 


2 


90.886066 


94.122602 


95.806853 


97.632698 


101.055836 


3 


92.113277 


95.436343 


97.166092 


99.044849 


102.561115 


4 


93.343556 


96.754463 


98.530428 


100.462884 


104.073921 


5 


94.576911 


98.076977 


99.899880 


101.886828 


105.594290 


6 


95.813349 


99.403899 


101.274465 


103.316705 


107.122261 


7 


97.052878 


100.735244 


102.654207 


104.752540 


108.657872 


8 


98.295506 


102.071027 


104.039123 


106.194357 


110.201161 


9 


99.541244 


103.411263 


105.429233 


107.642182 


111.752167 


90 


100.790094 


104.755966 


106.824556 


109.096040 


113.310928 


1 


102.042065 


106.105151 


108.225111 


110.555955 


114.877483 


2 


103.297166 


107.458834 


109.630919 


112.021953 


116.451870 


3 


104.555405 


108.817032 


111.041998 


113.494060 


118.034129 


4 


105.816790 


110.179754 


112.458769 


114.972300 


119.624300 


5 


107.081328 


111.547019 


113.880651 


116.456700 


121.222418 


6 


108.349027 


112.918841 


115.307661 


117.947285 


122.828530 


7 


109.619895 


114.295236 


116.740026 


119.444077 


123.442672 


8 


110.893941 


115.676219 


118.177762 


120.947109 


125.064885 


9 


112.171172 


117.061806 


119.620890 


122.456404 


126.695210 


100 


113.451596 


118.452111 


121.069429 


123.971992 


128.333686 


1 


114.735220 


119.846950 


122.523400 


12§.493891 


129.980354 


2 


116.022054 


121.246438 


123.982824 


127.022128 


131.635256 


3 


117.312105 


122.650591 


125.447718 


128.556936 


133.298432 


4 


118.605381 


124.059425 


1 126.918108 


130.097738 


134.979924 


5 


119.901890 


125.472955 


128.394012 


131.645161 


136.659774 


6 


121.201641 


126.891197 


1 129.875451 


133.199032 


138.338023 


7 


122.504641 


128.314167 


! 131.362445 


134.759377 


140.034713 


8 


123.810901 


129.741879 


I 132.855015 


[ 136.326224 


1 141.749887 


9 


125.120424 


131.174350 


| 134.353182 


1 137.899599 


143.453586 


110 


126.433221 


132.611597 


| 135.856969 


' 139.479530 


145.175854 


1 


127.749300 


134.053634 


| 137.366387 


1 141.066044 


146.906733 


2 


129.068669 


135.500478 


! 138.881472 


1 142.659166 


! 148.646267 


3 


130.391336 


136.952143 


I 140.402239 


! 144.259028 


150.394498 


4 


131.719310 


138.408645 


1 131.928708 


| 145.865456 


| 152.151471 


5 


133.046599 


139.870006 


1 143.460901 


! 147.478577 


' 153.917229 


6 


134.379211 


141.336241 


I 144.998840 


1 149.098420 


155.691815 


7 


135.715155 


142.807360 


1 146.542547 


150.725012 


157.475274 


8 


137.054439 


144.283383 


148.092042 


152.358382 


159.267651 


9 


138.397071 


1 145.764326 


1 149.647348 


1 153.998557 


| 161.068989 


120 


139.743059 


] 147.250206 


| 150.208487 


] 155.645187 


| 162.879334 



80 



FINANCE AND LIFE INSURANCE. 



TABLE NO. IV. Continued. 

The Amount of One Dollar Payable at the End of Each 
Period, Improved at Compound Interest at the Rate 

and for the Terms Stated. Snl 



Periods 


5-8% 


3-4% 


7-8% 


1% 


1 1-8 % 


1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


2.006250 


2.007500 


2.008750 


2.010000 


2.011250 


3 


3.018789 


3.022500 


3.026327 


3.030100 


3.033913 


4 


4.037654 


4.045190 


4.052807 


4.060401 


4.068044 


5 


5.062889 


5.075329 


5.088269 


5.101005 


5.113809 


6 


6.094532 


6.113596 


6.132791 


6.152015 


6.171340 


7 


7.132623 


7.159448 


7.186453 


7.213535 


7.240767 


8 


8.177199 


8.213146 


8.249335 


8.285671 


8.322225 


9 


9.228305 


9.274745 


9.321516 


9.368527 


9.415849 


10 


10.285984 


10.344281 


10.403080 


10.462213 


10.521776 


11 


11.350271 


11.421863 


11.494107 


11.566835 


11.640145 


12 


12.421210 


12.507529 


12.594680 


12.628503 


12.771096 


13 


13.498843 


13.601336 


13.704884 


13.809328 


13.914770 


14 


14.583211 


14.703346 


14.824801 


14.947421 


15.071310 


15 


15.674356 


15.813621 


15.954518 


16.096896 


16.240834 


16 


16.772321 


16.932224 


17.094120 


17.257864 


17.423543 


17 


17.877147 


18.059216 


18.243694 


18.430443 


18.619557 


18 


18.988879 


19.194660 


19.403326 


19.614748 


19.829024 


19 


20.107462 


20.338620 


20.573105 


20.810895 


21.052100 


20 


21.233134 


21.491160 


21.753120 


22.019004 


22.288935 


21 


22.365842 


22.652342 


22.943460 


23.239194 


23.539685 


22 


23.505629 


23.822335 


24.144215 


24.471586 


24.804506 


23 


24.652539 


25.001002 


25.355477 


25.716302 


26.078556 


24 


25.806618 


26.188573 


26.577337 


26.973465 


27.371995 


25 


26.967910 


27.384986 


27.809889 


28.243199 


28.679985 


26 


28.136460 


28.590372 


29.053226 


29.525632 


30.002690 


27 


29.312413 


29.804798 


30.307441 


30.820888 


31.340376 


28 


30.495615 


31.028304 


31.572631 


32.129097 


32.692909 


29 


31.686212 


32.261015 


32.848892 


33.450388 


34.060760 


30 


32.884250 


33.502971 


34.136320 


34.784892 


35.443999 


31 


34.099765 


34.754242 


35.435012 


36.132740 


36.842799 


32 


35.312825 


36.014892 


36.745069 


37.494068 


38.257336 


33 


36.533467 


37.285002 


38.066558 


38.869009 


39.687786 


34 


37.761738 


38.564628 


39.399671 


40.257699 


41.134329 


35 


38.997687 


39.853861 


40.744418 


41.660276 


42.597145 


36 


40.241360 


41.152764 


42.100932 


43.076878 


44.076418 


37 


41.492805 


42.461408 


43.469315 


44.507647 


45.572333 


38 


42.752072 


43.779864 


44.849671 


45.952724 


47.085077 


39 


44.019206 


45.108212 


46.242106 


47.412251 


48.614839 


40 


45.294260 


46.446522 


47.646724 


48.886373 


50.161811 


41 


46.577286 


47.794870 


49.063633 


50.375237 


51.726187 


42 


47.868331 


49.153330 


50.492940 


51.878989 


53.308162 


43 


49.167445 


50.521979 


51.934753 


53.397779 


54.907934 


44 


50.474678 


51.900892 


53.389182 


54.931757 


56.525704 


45 


51.790082 


53.290147 


54.856338 


56.481075 


58.161673 


46 


53.113717 


54.689822 


56.336331 


58.045885 


59.816047 


47 


54.445614 


56.099994 


57.829273 


59.626344 


' 61.489033 


48 


55.785836 


57.520743 


59.335280 


61.222608 


63.180840 


49 


57.134434 


58.952147 


60.854463 


62.834834 


64.891767 


50 


58.491458 


60.394287 


62.386940 


64.463182 


66.621767 


51 


59.856966 


61.847243 


63.932826 


66.107814 


68.371317 


52 


61.231019 


63.311096 


65.492238 


67.768892 


70.140549 


53 


62.613649 


64.785944 


67.065296 


69.446582 


72.649686 


54 


64.004923 


66.271805 


68.652118 


71.141048 


73.738950 


55 


65.404891 


67.768857 


70.252825 


72.852459 


I 75.568564 


56 


66.813608 


69.277123 


71.867538 


74.580984 


77.418766 


57 


68.231129 


,70.796700 


| 73.496379 


I 76.326794 


| 79.289782 


58 


69.657510 


72.327674 


| 75.139472 


| 78.090062 


81.181847 


59 


71.092806 


73.870130 


76.796943 


| 79.870963 


| 83.095198 


60 


72.537073 


75.424154 


1 78.468916 


| 81.669674 


j 85.030074 



ACCUMULATED ANNUITIES. 



81 



TABLE NO. IV, Continued. 

The Amount of One Dollar Payable at the End of Each 
Period, Improved at Compound Interest at the Rate 

and for the Terms Stated. Sn| 



Periods 


1 1-4 % 


1 3-8 % 


1 1-2 % 


1 5-8 % 


1 3-4 % 


1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


2.012500 


2.013750 


2.015000 


2.016250 


2.017500 


3 


3.037656 


3.041438 


3.045225 


3.049014 


3.052806 


4 


4.075626 


4.083257 


4.090903 


4.098561 


4.106230 


5 


5.126572 


5.139401 


5.152267 


5.165162 


5.178089 


6 


6.190654 


6.212536 


6.229551 


6.249096 


6.268706 


7 


7.268037 


7.297924 


7.322994 


7.350644 


7.378408 


8 


8.358988 


8.393236 


8.432839 


8.470092 


8.507530 


9 


9.463474 


9.513678 


9.559332 


9.607731 


9.656412 


10 


10.581766 


10.644457 


10.702722 


10.763856 


10.825399 


11 


11.714037 


11.790779 


11.863262 


11.938770 


12.014844 


12 


12.860461 


12.952868 


13.041211 


13.132775 


13.225104 


13 


14.021215 


14.130936 


14.236830 


14.346182 


14.456543 


14 


15.196479 


15.325200 


15.450382 


15.579308 


15.709532 


15 


16.386433 


16.535887 


16.682138 


16.832472 


16.984449 


16 


17.591262 


17.763222 


17.932370 


18.106000 


18.281677 


17 


18.811151 


19.007432 


19.201355 


19.400223 


19.601607 


18 


20.046288 


20.268750 


20.489375 


20.715477 


20.944635 


19 


21.296865 


21.547411 


21.796716 


22.052103 


22.311166 


20 


22.563074 


22.843654 


23.123667 


23.410450 


23.701611 


21 


23.845111 


24.157720 


24.470522 


24.790870 


25.116389 


22 


25.143173 


25.489855 


25.837580 


26.193689 


26.555926 


23 


26.460491 


26.840307 


27.225144 


27.619337 


28.020655 


24 


27.791207 


28.209327 


28.633521 


29.068152 


29.511016 


25 


29.138558 


29.597171 


30.063024 


30.540510 


31.027459 


26 


30.502751 


31.004098 


31.513969 


32.036794 


32.570440 


27 


31.883996 


32.430370 


32.986678 


33.557393 


34.140422 


28 


33.282507 


33.879587 


34.481479 


35.102705 


35.737880 


29 


34.698499 


35.345352 


35.998701 


36.673124 


37.363293 


30 


36.132191 


36.831271 


37.538681 


38.269063 


39.017150 


31 


37.583804 


38.337621 


39.101762 


39.890936 


40.699950 


32 


39.153562 


39.864684 


40.688288 


41.539164 


42.412199 


33 


40.541692 


41.412744 


42.298612 


43.214176 


44.154413 


34 


42.051897 


42.982090 


43.933091 


44.916407 


45.927115 


35 


43.577463 


44.573014 


45.592088 


46.646299 


47.730840 


36 


45.122099 


46.185814 


47.275969 


48.404302 


49.566129 


37 


46.689648 


47.820787 


48.985109 


50.190873 


51.433537 


38 


48.273140 


49.478243 


50.719885 


52.006475 


53.333624 


39 


49.876426 


51.158489 


52.480684 


53.851581 


55.266692 


40 


51.499754 


52.861840 


54.267894 


55.726670 


57.234134 


41 


53.143373 


54.588611 


56.081912 


57.632229 


59.235731 


42 


54.807537 


56.339125 


57.923141 


59.568753 


61.272357 


43 


56.492504 


58.113708 


59.791988 


61.536746 


63.344623 


44 


58.198533 


59.912692 


61.688868 


63.536721 


65.453154 


45 


59.925887 


61.736412 


63.614201 


65.569193 


67.598584 


46 


61.696694 


63.585209 


65.568414 


67.634693 


69.781559 


47 


63.467501 


65.459426 


67.551940 


69.733757 


72.002736 


48 


65.260443 


67.359414 


69.565219 


71.866931 


74.262784 


49 


67.075786 


69.285527 


71.608697 


74.034769 


76.562383 


50 


68.313832 


71.238123 


73.682828 


76.237835 


78.902225 


51 


70.774853 


73.217568 


75.788070 


78.476700 


81.283014 


52 


72.659148 


75.224231 


77.924891 


80.751947 


83.705466 


53 


74.566986 


77.258485 


80.093764 


83.064167 


86.170311 


54 


76.498672 


79.320710 


82.295170 


85.413960 


88.678291 


55 


78.454505 


81.411291 


84.529597 


87.801937 


91.230154 


56 


80.434785 


83.530618 


86.797541 


90.228719 


93.826681 


57 


82.439819 


35.679085 


89.099503 


92.694936 


96.468647 


58 


84.469916 


87.857093 


91.436996 


95.201228 


99.156847 


59 


86.525388 


90.065049 


93.807536 


97.748249 


101.892191 


60 


88.606554 


92.303364 


96.214649 


100.336658 


104.675301 



82 



FINANCE AND LIFE INSURANCE. 



TABLE NO. IV. Continued. 

The Amount of One Dollar Payable at the End of Each 
Period, Improved at Compound Interest at the Rate 

and for the Terms Stated. Snl 



Periods 



1 7-8 % 



2% 



2 1-4 % 



2 1-2 % 



2 3-4 % 



1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


2.018750 


2.020000 


2.022500 


2.025000 


2.027500 


3 


3.056601 


3.060400 


3.068006 


3.075625 


3.083256 


4 


4.113912 


4.121608 


4.137036 


4.152516 


4.168046 


5 


5.191048 


5.204040 


5.230120 


5.256329 


5.282667 


6 


6.288380 


6.308121 


6.347797 


6.387737 


6.427940 


7 


7.406347 


7.434283 


7.490622 


7.547430 


7.604709 


8 


8.545214 


8.582969 


8.659162 


8.736116 


8.813838 


9 


9.705435 


9.754628 


9.853993 


9.954519 


10.056219 


10 


10.887210 


10.949721 


11.075708 


11.203382 


11.332765 


11 


12.091347 


12.168715 


12.324911 


12.483466 


12.644416 


12 


13.318062 


13.412090 


13.602222 


13.795553 


13.992137 


13 


14.567777 


14.680332 


14.908272 


15.140442 


15.376921 


14 


15.840925 


15.973938 


16.243708 


16.518953 


16.799786 


15 


17.137944 


17.293417 


17.609191 


17.931927 


18.261781 


16 


18.459282 


18.639285 


19.005398 


19.380225 


19.763979 


17 


19.805395 


20.012071 


20.433020 


20.864730 


21.307489 


18 


21.176784 


21.412312 


21.892762 


22.386349 


22.893445 


19 


22.574031 


22.840559 


23.385350 


23.946007 


24.523015 


20 


23.997292 


24.297370 


24.911520 


25.544658 


26.197398 


21 


25.447238 


25.783317 


26.472029 


27.183274 


27.917826 


22 


26.924371 


27.298984 


28.067650 


28.862856 


29.685566 


23 


28.429200 


28.844963 


29.699172 


30.584427 


31.501919 


24 


29.962244 


30.421862 


31.367403 


32.349038 


33.368222 


25 


31.524034 


32.030300 


33.073170 


34.157764 


35.285848 


26 


33.115106 


33.670906 


34.817316 


36.011708 


37.256209 


27 


34.736011 


35.344324 


36.600706 


37.912001 


39.280755 


28 


36.387308 


37.051210 


38.424222 


39.859801 


41.360975 


29 


38.069567 


38.792234 


40.288767 


41.856296 


43.498402 


30 


39.783367 


40.568079 


42.195264 


43.902703 


45.694608 


31 


41.529298 


42.379441 


44.144657 


46.000271 


47.951210 


32 


43.307969 


44.227030 


46.137912 


48.150278 


50.269868 


33 


45.119990 


46.111570 


48.176015 


50.354034 


52.652290 


34 


46.965986 


48.033802 


50.259976 


52.612885 


55.100228 


35 


48.846673 


49.994478 


52.390825 


54.928207 


57.615484 


36 


50.762543 


51.994367 


54.569619 


57.301413 


60.199910 


37 


52.713886 


54.034254 


56.797435 


59.733948 


62.855407 


38 


54.702274 


56.114940 


59.075377 


62.227297 


65.583930 


39 


56.727945 


58.237238 


61.404573 


64.782979 


68.387489 


40 


58.791592 


60.401983 


63.786176 


67.402553 


71.268145 


41 


60.893937 


62.610023 


66.221365 


70.087617 


74.228019 


42 


63.035701 


64.862223 


68.711346 


72.839808 


77.269289 


43 


65.217623 


67.159468 


71.257351 


75.660803 


80.394195 


44 


67.440456 


69.502657 


73.860642 


78.552323 


83.605035 


45 


69.704967 


71.892710 


76.522506 


81.516131 


86.904174 


46 


72.011927 


74.330564 


79.244262 


84.554034 


90.294039 


47 


74.362153 


76.817176 


82.027258 


87.667885 


93.777125 


48 


76.756445 


79.353519 


84.872872 


90.859582 


I 97.355996 


49 


79.195630 


1 81.940590 


1 87.782511 


94.131072 


I 101.033285 


50 


81.680550 


| 84.579401 


| 90.757618 


97.484349 


I 104.811701 


51 


84.212062 


I 87.270989 


I 93.799664 


100.921457 


| 108.694022 


52 


86.791040 


90.016409 


96.910166 


104.444493 


I 112.683108 


53 


89.418371 


92.816737 


| 100.090644 


108.055605 


| 116.781893 


54 


92.094966 


95.673071 


I 103.342683 


111.756995 


| 120.993395 


55 


94.821747 


1 98.586532 


| 106.667893 


115.550920 


1 125.320713 


56 


97.599736 


1 101.558262 


1 110.067920 


1 119.439693 


1 129.767032 


57 


I 100.429666. 


I 104.589427 


1 113.544448 


I 123.425685 


I 134.335625 


58 


103.312923 


1 107.681215 


1 117.099198 


I 127.511327 


I 139.029856 


59 


1 106.249837 


1 110.834839 


I 120.733930 


[ 131.699110 


I 143853176 


60 


| 109.242022 


| 114.051535 


| 124.450433 


| 135.991588 


J 148.809138 



ACCUMULATED ANNUITIES. 



83 



TABLE XO. IV. Continue!. 

The Amount of One Dollar Payable at the End of Each 
Period, Improved at Compound Interest at the Rate 

and for the Terms Stated. Sni 



Periods 


3% 


3 1-2 % 


4% 


4 1-2 % 


5% 


1 


1.00000 


1.00000 


1.00000 


1.00000 | 


1.00000 


2 


2.03000 


2.03500 


2.04000 


2.04500 | 


2.05000 


3 


3.09090 


3.10623 


3.12160 


3.13703 | 


3.15250 


4 


4.18363 


4.21494 


4.24646 


4.27819 | 


4.31013 


5 


5.30914 


5.56247 


5.41632 


5.47071 1 


5.52563 


6 


6.46841 


6.55015 


6.63298 


6.71689 | 


6.80191 


7 


7.66246 


7.77941 


7.89829 


8.01915 | 


8.14201 


8 


8.89234 


9.05169 


9.21423 


9.38001 | 


9.54911 


9 


10.15911 


10.36850 


10.58280 


10.80211 | 


11.02656 


10 


11.46388 


11.73139 


12.00611 


12.28S21 | 


12.57789 


11 


12.80780 


13.14199 


13.48635 


13.84118 | 


14.20679 


12 


14.19203 


14.60196 


15.02581 


15.46403 | 


15.91713 


13 


15.61779 


16.11303 


16.62684 


17.15991 | 


17.71298 


14 


17.08632 


17.67699 


18.29191 


17.93211 | 


19.59863 


15 


18.59891 


19.29568 


20.02359 


20.78405 | 


21.57856 


16 


20.15688 


20.97103 


21.82453 


22.71934 | 


23.65749 


17 


21.76159 


22.70502 


23.69751 


24.74171 | 


25.84037 


18 


23.41444 


24.49969 


25.64541 


26.85508 


28.13238 


19 


25.11687 


26.35718 


27.67123 


29.06356 | 


30.53900 


20 


26.87037 


28.27968 


29.77808 


31.37142 | 


33.06595 


21 


28.67649 


30.26947 


31.96920 


33.78314 | 


35.71925 


22 


30.53678 


32.32890 


34.24797 


36.30338 I 


38.50521 


23 


32.45288 


34.46041 


36.61789 


38.93703 | 


41.43048 


24 


34.42647 


36.66653 


39.08260 


41.68920 | 


44.50200 


25 


36.45926 


39.94986 


41.64591 


44.56521 | 


47.72710 


26 


38.55304 


41.31310 


44.31174 


47.57064 | 


51.11345 


27 


40.70963 


43.75906 


47.08421 


50.71132 


54.66913 


28 


42.93092 


46.29063 


49.96758 


53.99333 | 


58.40258 


29 


45.21885 


48.91080 


52.96629 


57.42303 | 


62.32271 


30 


47.57542 


51.62268 


56.08494 


61.00707 | 


66.43885 


31 


50.00268 


54.42947 


59.32834 


64.75239 | 


70.76079 


32 


52.50276 


57.33450 


62.70147 


68.66625 | 


75.29883 


33 


55.07784 


60.34121 


66.20953 


72.75623 | 


80.06377 


34 


57.73018 


63.45315 


69.85791 


77.03026 | 


85.06696 


35 


60.46208 


66.67401 


73.65222 


81.49662 I 


90.32031 


36 


63.27594 


70.00760 


77.59831 


86.16397 | 


95.83632 


37 


66.17422 


73.45787 


81.70225 


91.04134 | 


101.62814 


38 


69.15945 


77.02889 


85.97034 


96.13820 | 


107.70955 


39 


72.23423 


80.72491 


90.40915 


101.46442 | 


114.09502 


40 


75.40126 


84.55028 


95.02552 


107.03032 | 


120.79977 


41 


78.66330 


88.50954 


99.82654 


112.84669 1 


127.83976 


42 


82.02320 


92.60737 


104.81960 


118.92479 | 


135.23175 


43 


85.48389 


96.84863 


110.01238 


125.27640 1 


142.99334 


44 


89.04841 


101.23833 


115.41288 


131.91384 | 


151.14301 


45 


92.71986 


105.78167 


121.02939 


138.84997 | 


159.70016 


46 


96.50146 


110.48403 


126.87057 


146.09821 


168.68516 


47 


100.39650 


115.35097 


132.94539 


153.67263 1 


178.11942 


48 


104.40840 


120.38826 


139.26321 


161.58790 | 


188.02539 


49 


108.54065 


125.60185 


145.83373 


169.85936 1 


198.42660 


50 


112.79687 


130.99791 


152.66708 


178.50303 1 


209.34800 


51 


117.18077 


136.58284 


159.77377 


187.53566 1 


220.81540 


52 


121.69620 


142.36324 


167.16472 


196.97477 ! 


232.85617 


53 


126.34708 


1 148.34595 


174.85131 


206.83863 1 


245.49879 


54 


131.13749 


1 154.53806 


182.84536 


217.14637 1 


258.77392 


55 


136.07162 


1 160.94689 


181.15917 


227.91796 1 


272.71262 


56 


141.15377 


1 167.58003 


1 199.80544 


239.17427 


287.34825 


57 


146.38838 


! 174.44533 


208.79776 


250.93711 "1 


302.71566 


58 


151.78003 


1 181.55092 


I 218.14967 


263.22928 


318.85144 


59 


157.33343 


1 188.90520 


1 227.87566 


276.07460 1 


335.79402 


60 


163.05344 


( 196.51688 


237.99069 


289.49795 | 


353.58372 



84 



FINANCE AND LIFE INSURANCE. 



TABLE NO. IV. Concluded. 

The Amount of One Dollar Payable at the End of Each 
Period, Improved at Compound Interest at the Rate 

and for the Terms Stated. Snl 



Periods 


6% 


7% 


8% 


»% 


10% 


1 


1.00000 


1.00000 


1.00000 


| 1.00000 


1.00000 


2 


2.06000 


2.07000 


2.08000 


2.09000 


2.10000 


3 


3.18360 


3.21490 


3.24640 


3.27810 


3.31000 


4 


4.37462 


4.43994 


4.50611 


4.57312 


4.64100 


5 


5.63709 


5v75074 


5.86660 


5.98470 


6.10510 


6 


6.97532 


7.15329 


7.33593 


7.52332 


7.71561 


7 


8.39384 


8.65402 


8.92280 


9.20042 


9.48717 


8 


9.89747 


10.25980 


10.63662 


11.02846 


11.43589 


9 


11.49132 


11.97799 


12.48755 


13.02102 


13.57948 


' 10 


13.18079 


13.81645 


14.48655 


15.19291 


15.93743 


11 


14.97164 


15.78360 


16.64548 


17.56027 


18.53117 


12 


16.86994 


17.88845 


18.97712 


20.14070 


21.38429 


13 


18.88214 


20.14067 


21.49529 


22.95336 


24.52272 


14 


21.01507 


22.55052 


24.21491 


.26.01916 


27.97499 


15 


23.27597 


25.12905 


27.15210 


29.36087 


31.77249 


16 


25.67253 


27.88808 


30.32427 


33.00337 


35.94974 


17 


28.21288 


30.84024 


33.75021 


36.97367 


40.54471 


18 


30.90565 


33.99906 


37.45023 


41.30130 


45.59918 


19 


33.75999 


37.37899 


41.44625 


46.01842 


51.15910 


20 


36.78559 


40.99552 


45.76195 


51.16008 


57.27501 


21 


39.99273 


44.86520 


50.42291 


56.76449 


64.00251 


22 


43.39229 


49.00576 


55.45674 


62.87330 


71.40276 


23 


46.99583 


53.43616 


60.89328 


69.53190 


79.54303 


24 


50.81538 


38.17669 


66.76474 


76.78977 


88.49733 


25 


54.86451 


63.24906 


73.10592 


84.70085 


98.34706 


26 


59.15638 


68.67649 


79.95440 


93.32393 


109.18177 


27 


63.70577 


74.48384 


87.35075 


102.72309 


121.09995 


28 


68.52811 


80.69771 


95.33881 


112.96817 


134.20994 


29 


73.63980 


87.34655 


103.96592 


124.13531 


148.63093 


30 


79.05819 


94.46081 


113.28319 


136.30749 


164.49402 


31 


84.80168 


102.07307 


123.34585 


149.57517 


181.94342 


32 


90.88978 


110.21818 


134.21352 


164.03694 


201.13776 


33 


97.34316 


118.93345 


145.95060 


179.80027 


222.25154 


34 


104.18375 


128.25879 


158.62665 


196.98230 


245.47669 


35 


111.43478 


138.23690 


172.31678 


215.71071 


271.02436 


36 


119.12087 


148.91348 


187.10212 


236.12468 


299.12680 


37 


127.26812 


160.33742 


203.07029 


258.37591 


330.03948 


38 


135.90421 


172.56104 


220.31592 


282.62975 


364.04344 


39 


145.05846 


185.64031 


238.94120 


309.06643 


401.44773 


40 


154.76197 


199.63513 


259.05650 


337.88241 


442.5925G 


41 


165.04768 


214.60959 


280.78102 


369.29183 


487.85182 


42 


175.95054 


230.63226 


304.24350 


403.52810 


537.63700 


43 


187.50758 


247.77652 


329.58298 


440.84563 


592.40070 


44 


199.75803 


266.12087 


356.94962 


481.52174 


652.64077 


45 


212.74351 


285.74933 


386.50559 


525.85870 


718.90485 


46 


226.50812 


306.75178 


418.42604 


574.18599 


791.79533 


47 


241.09861 


329.22440 


452.90013 


626.86273 


871.97486 


48 


256.56453 


353.27011 


490.13214 


684.28038 


960.17235 


49 


272.95840 


378.99902 


530.34271 


746.86562 


1057.18958 


50 


290.33590 


406.52895 


573.77013 


815.08353 


1163.90854 


51 


308.75606 


435.98598 


620.67174 


889.44105 


1281.29939 


52 


328.28142. 


467.50500 


671.32548 


970.49074 


1410.42933 


53 


348.97831 


501.23035 


726.03152 


1058.83491 


1552.47226 


54 


370.91701 


537.31647 


785.11404 


1155.13006 


1708.71949 


55 


394.17203 


575.92862 


848.92317 


1260.09177 


1880.59144 


56 


418.82235 


617.24362 


917.83703 


1374.50003 


2069.65058 


57 


444.95169. 


661.45067 


992.26400 


1499.20503 


2277.61563 


58 


472.64879 


708.75221 


1072.64513 


1635.13348 


2506.37719 


59 


502.00772 


759.36485 


1159.45685 


1783.29549 


2758.01491 


60 


553.12818 


813.52038 


1253.21351 


1944.49208 


3034.81640 



86 



FINANCE AND LIFE INSURANCE. 



TABLE NO. V. 

The Present Value of One Dollar Payable at the End of 

Each Period at the Rates and for the Terms Stated. an\ 



Periods 



1-4% 



1-3 



3-8 



5-12% 



1-2% 



1 


.997506 


.996678 | 


.996264 


.995851 | 


.995025 


2 


1.992525 


1.990044 1 


1.988883 


1.987569 j 


1.985100 


3 


2.985062 


2.980050 | 


2.977717 


2.975172 


2.970249 


4 


3.975124 


3.966827 


3.962857 


3.958677 | 


3.950497 


5 


4.962717 


4.950326 | 


4.944314 


4.938102 


4.925867 


6 


5.947847 


5.930557 | 


5.922107 


5.913463 | 


5.896385 


7 


6.930521 


6.907532 1 


6.896247 


6.884776 


6.862075 


8 


7.910744 


7.881261 | 


7.866681 


7.852059 


7.822960 


9 


8.888522 


8.851755 


8.833556 


8.815324 | 


8.779065 


10 


9.863862 


9.819025 


9.796819 


9.774596 


9.730413 


11 i 


10.836768 


10.783081 1 


10.756481 


10.729888 


10.b77028 


12 


11.807549 


11.743936 


11.712561 


11.681216 | 


11.618933 


13 | 


12.775610 


12.701596 1 


12.665069 


12.628597 | 


12.556152 


14 | 


13.741257 


13.656075 


13.614018 


13.572046 


13.488708 


15 | 


14.704496 


14.607383 


14.559420 


14.511581 


14.416626 


16 


15.665333 


15.555531 


15.501292 


15.447217 


15.339925 


17 


16.623774 


16.500529 


16.439646 


16.378971 


16.258632 


18 


17.579824 


17.442387 


17.374494 


17.306859 


17.172768 


19 


18.533490 


18.381116 


18.406049 


18.230897 


18.082356 


20 


19.484778 


19.316727 


19.233925 


19.151101 


18.987419 


21 


20.433694 


20.249229 


20.158334 


20.067487 


19.887979 


22 


21.380248 


21.178633 


21.079290 


20.980070 


20.784059 


23 


22.324432 


22.104950 


21.996806 


21.888863 


21.675681 


24 


23.266266 


23.028189 


22.910894 


22.793888 


22.562867 


25 


24.205751 


23.948361 


23.821567 


23.695158 


23.445639 


26 


25.142893 


24.865476 


24.728838 


24.592688 


24.324019 


27 


26.077701 


25.779544 


25.632719 


25.486494 


25.198029 


28 


27.010175 


26.690575 


26.533224 


26.376592 


26.067691 


29 


27.940322 


27-598580 


27.430365 


27.262996 


26.933026 


30 


28.868149 


28.503586 


28.324154 


28.145722 


27.794056 


31 


29.793366 


29.405549 


29.214604 


29.024785 


28.650802 


32 


30.716883 


30.304534 


30.101727 


29.900201 


29.503286 


33 


31.637786 


31.200532 


30.985536 


30.771984 


30.351528 


34 


32.556394 


32.093553 


31.866043 


31.640150 


31.195550 


35 


33.472712 


32.983607 


32.743261 


32.504714 


32.035373 


36 


34.386744 


33.870704 


33.617209 


33.365690 


32.871018 


37 


35.298497 


34.754854 


34.487885 


34.223094 


33.702505 


38 


36.207976 


35.636067 


35.355308 


35.076940 


34.529856 


39 


37.115187 


36.514352 


36.219491 


35.927243 


35.353091 


40 


38.020126 


37.389719 


37.080445 


36.774018 


36.172230 


41 


38.922818 


38.262178 


37.938183 


37.617279 


36.987294 


42 


39.823259 


39.131739 


38.792716 


38.457041 


37.798303 


43 


40.721454 


39.998411 


39.644057 


39.293319 


38.605277 


44 


41.617409 


40.862203 


40.492218 


40.126127 


39.408236 


45 


42.511130 


41.723126 


41.337210 


40.955479 


40.207200 


46 


43.402622 


42.581188 


42.179045 


41.781390 


41.002187 


47 


44.291891 


43.436400 


43.017735 


42.603874 


41.793223 


48 


45.178942 


44.288771 


43.853291 


43.422945 


42.579321 


49 


46.063781 


45.138310 


44.685726 


44.238617 


43.363504 


50 


46.946413 


45.985026 


45.515052 


45.050904 


44.142790 


51 


47.826847 


46.828929 


46.341279 


45.859821 


44.918199 


52 


48.705083 


47.670028 


47.164410 


46.665382 


45.689750 


53 


49.581129 


48.508333 


47.984476 


47.467600 


46.457463 


54 


50.454990 


49.343853 


48.801478 


48.266490 


47.221356 


55 


51.326672 


50.176597 


49.615428 


49.062064 


47.981448 


56 


52.196180 


51.006575 


I 50.426337 


49.854337 


48.737759 


57 


53.063520 


51.833795 


| 51.234216 


I 50.643323 


49.490308 


58 


53.928697 


52.658267 


1 52.039077 


j 51.429035 


50.239112 


59 


54.791716 


' 53.480000 


| 52.840932 


I 52^.211488 


50.984192 


6a 


55.652583 


54.299003 


| 53.639791 


| 52.990695 


51.725564 



ANNUITIES CERTAIN. 



S7 



TABLE XO. V. Continued. 
The Present Value of One Dollar Payable at the End of 
Each Period at the Rates and for the Terms Stated. <zn| 



Periods 


1-4% 


1-3 % 


3-8% 


5-12 % 


1-2% 


61 | 


56.511305 1 


55.115287 | 


54.435656 j 


53.766667 


52.463248 


62 | 


57.367886 


55.92S857 


55.228551 


54.539419 


53.197262 


63 | 


58.222330 , 


56.739724 


56.018483 


55.308965 


53.927624 


64 


59.074644 


57.547897 


56.805464 | 


56.075318 


54.654352 


65 


59.924832 


58.353385 j 


57.589505 


56.838491 


55.377465 


66 


60.727900 


59.156197 


58.370621 


57.598497 


56.096980 


67 


61.618853 


59.956342 | 


59.148815 


58.355350 


56.812915 


68 


62.461697 j 


60.753829 


59.924101 


59.109062 


57.525289 


69 


63.303436 


61.54S667 


60.696491 


59.859647 


58.234118 


70 | 


64.143076 | 


62.340861 


61.465995 


60.607117 


58.939421 


71 | 


64.980622 


63.130426 


62.232624 


61.351486 


59.641215 


72 


65.816080 | 


63.917368 


62.996389 


62.092766 


60.339517 


73 


66.649455 


64.701695 


63.757301 


62.830970 


61.034345 


74 


67.470751 


65.483417 


64.515370 


63.566111 


61.725716 


75 


68.299974 


66.262542 


65.270670 


64.298202 


62.413648 


76 


69.127129 


67.039078 


66.023022 


65.027255 


63.098157 


77 


69.952222 


67.813034 


66.772626 


65.753283 


63.779261 


78 


70.775257 


68.584422 


67.519430 


66.476298 


64.456976 


79 


71.596239 


69.353244 


68.263444 


67.196313 


65.131319 


80 


72.415174 


70.119512 


69.004678 


67.913337 


65.802307 


81 | 


73.232067 


70.883234 


69.743143 


68.627390 


66.469957 


82 | 


74.046913 


71.644419 


70.478849 


69.338480 


67.134286 


83 | 


. 74.859737 


72.403075 


71.211806 


70.046619 


67.795309 


84 | 


75.670531 


73.159211 


71.942125 


70.751820 


68.453044 


85 | 


76.479306 


73.912835 


72.669616 


71.454095 


69.107506 


86 


77.285321 


74.663955 


73.394389 


72.153456 


69.758712 


87 I 


78.089067 


75.412580 


74.116454 


72.849915 


70.406679 


88 


78.891806 


76.158717 


74.835822 


73.543484 


71.051422 


89 


79.692548 


76.892376 


75.552502 


74.234275 


71.692957 


90 


80.491284 


77.633564 


76.266505 


74.922100 


72.331300 


91 


81.288033 


78.372289 


76.977840 


75.607071 


72.966468 


92 


82.082795 


79.108460 


77.686517 


76.289200 


73.598476 


93 


82.875574 


99.842285 


78.392547 


76.968498 


74.227339 


94 


83.666377 


80.573672 


79.095939 


77.644978 


74.853074 


95 


84.455218 


81.302629 


79.796703 


78.318651 


75.475695 


96 


85.242081 


82.029165 


80.494849 


78.989528 


76.095219 


97 


86.026982 


82.753287 


81.190387 


79.657622 


76.711661 


98 


86.809926 


83.475003 


81.883326 


80.322942 


77.325036 


99 


87.590917 


84.194323 


82.573677 


80.985503 


77.935359 


100 


88.369961 


84.911252 


83.261449 


81.644979 


78.542646 


101 


89.147062 


85.625799 


83.946651 


82.301053 


79.146911 


102 


89.922225 


86.337972 


84.529293 


82.955400 


79.748170 


103 


90.695455 


87.047779 


85.209385 


83.607032 


80.346438 


104 


91.466757 


87.755229 


85.886936 


84.255960 


80.941729 


105 


92.236135 


88.460327 


86.561955 


84.902196 


81.534059 


106 


93.003595 


89.163083 


87.234453 


85.545750 


82.123442 


107 


93.769141 


89.863504 


87.904439 


86.186634 


82.709893 


108 


94.532778 


90.561598 


88.571921 


86.824859 


83.293426 


109 


95.294510 


91.257373 


89.236910 


87.460435 


83.874056 


110 


96.054343 


91.950836 


89.899414 


88.093374 


84.451797 


111 


96.812280 


92.641995 


90.559443 


88.823687 


85.026664 


112 


97.568328 


93.330858 


91.217006 


89.341385 


85.598671 


113 


98.322490 


94.017433 


91.872113 


89.966478 


86.167832 


114 


99.074771 


94.701727 


92.524772 


90.588977 


86.734161 


115 


99.825177 


95.383747 


93.174993 


91.208893 


87.297673 


116 


100.573711 


96.063501 


93.822783 


91.826237 


87.858381 


117 


101.320379 


96.740997 


94.468156 


92.441019 


88.416300 


118 


102.065185 


97.416242 


95.111116 


93.053250 


88.971443 


119 


102.808133 


98.089244 


95.751718 


93.662941 


89.523824 


120 


103.549229 


98.760012 


96.389883 


94.270102 


90.083457 



FINANCE AND LIFE INSURANCE. 



TABLE NO. V. Continued. 
The Present Value of One Dollar Payable at the End of 



Each Period at the Rates and for the Terms Stated- 



an\ J, 



Periods 


5-8% 


3-4% 


7-8% 


1% 


1 1-8 % 


1 


| .99378 , 9 | .992556 


| .991326 


| .990099 


I .988875 


2 


| 1.981405 | 1.977723 


| 1.974053 


| 1.970395 


| 1.966749 


3 


2.962887 | 2.955557 


| 2.948256 


| 2.940985 


| 2.933744 


4 


| 3.938274 


3.926108 


| 3.914008 


| 3.901966 


| 3.889982 


5 


| 4.907601 


4.889438 


4.871381 


| 4.853431 


| 4.835582 


6 


| 5.870908 


5.845597 


| 5.820452 


5.795476 


| 5.770662 


7 


6.828232 


6.794638 


6.761291 


| 6.728195 


| 6.695340 


8 


| 7.779609 


7.736614 


| 7.693969 


7.651678 


| 7.609731 


9 


| 8.725077 


8.671578 


| 8.618557 


8.566018 


| 8.513950 


10 


| 9.664673 


9.599582 


| 9.535125 


| 9.471305 


| 9.408109 


11 


| 10.598432 


10.520678 


| 10.443742 


| 10.367628 


| 10.292323 


12 


| 11.526392 


11.434917 


| 11.344478 


| 11.255074 


| 11.166699 


13 


| 12.448588 


12.342351 


| 12.237401 


| 12.133740 


| 12.031347 


14 


| 13.365056 


13.243030 


| 13.122578 


13.003703 


12.886376 


15 


| 14.275916 


14.137004 


| 14.000077 


1 13.865053 


| 13.731893 


16 


| 15.181035 


15.024323 


| 14.869965 


| 14.717874 


14.568004 


17 


| 16.080532 


15.905037 


| 15.732307 


| 15.562251 


15.394813 


18 


| 16.974442 


16.779195 


| 16.587169 


| 16.398269 


16.212424 


19 


| 17.862801 


17.646846 


| 17.434616 


| 17.226009 


17.020940 


20 


1 18.755641 


18.508036 


18.274712 


| 18.045553 


17.820406 


21 


| 19.622997 


19.362817 


| 19.107521 


18.856983 


[ 18.611033 


22 


| 20.494904 


20.211235 


[ 19.933106 


| 19.660379 


19.392864 


23 


| 21.361395 


21.053337 


20.751530 


| 20.455821 


20.165998 


24 


22.222505 


21.889171 


21.562855 


21.243387 


20.930521 


25 


23.078275 


22.718783 


22.367142 


22.023156 


21.686548 


26 


23.928421 


23.542219 


23.164453 


22.795204 


22.434165 


27 


24.773585 


24.359526 


23.954848 


23.559608 


23.173464 


28 


25.613499 


25.170749 


24.738387 


24.316443 


23.904540 


29 


26.448196 


25.975933 


25.515130 


25.065785 


24.627482 


30 


27.277708 


26.775123 


26.285135 


25.807708 


25.342371 


31 


28.102068 


27.568364 


27.048461 


26.542285 


26.049317 


32 


28.921308 


28.355700 


27.805166 


27.269589 


26.748400 


33 


29.735460 


29.137175 


28.555307 


27.989693 


27.439705 


34 


30.544555 


29.912832 


29.298941 


28.702666 


28.123329 


35 


31.348624 


30.682716 


30.036125 


29.408580 


28.799338 


36 


32.147699 


31.446868 


30.766914 


30.107505 


29.467826 


37 


32.941811 


32.205330 


31.491365 


30.799510 


30.128878 


38 


33.730990 


32.958148 


32.209532 


31.484663 


30.782576 


39 


34.515268 


33.705360 


32.921469 


32.163033 


31.429001 


40 


35.294674 


34.447012 


33.627231 


32.834686 


32.068235 


41 


36.069238 


35.183143 


34.326871 


33.499689 


32.700358 


42 


36.838993 


35.913794 


35.020442 


34.158108 


33.325449 


43 


37.603966 


36.639006 


35.707997 


34.810008 


33.943586 


44 


38.364188 


37.358820 


36.389588 


35.455454 


34.554846 


45 


39.119688 


38.073275 


37.065267 


36.094508 


35.159306 


46 


39.870495 


38.782412 


37.735085 


36.727236 


35.757041 


47 


40.616639 


39.486270 


38.399093 


37.353699 


36.348127 


48 


41.358149 


40.184888 


39.057341 


37.937959 


36.932637 


49 


42.095053 


40.878306 


39.709880 


38.588079 | 


37.510644 


50 


42.827380 


41.566562 


40.356758 


39.196118 | 


38.082221 


51 


43.555158 


42.249695 


40.998928 


39.798136 | 


38.644440 


52 


44.278416 


42.927742 


41.634635 


40.394193 | 


39.206371 


53 


44.997182 


43.600741 


42.264828 


40.984349 | 


39.759084 


54 


45.711483 


44.268731 


42.889553 


41.568662 | 


40.305648 


55 


46.421357 


44.931749 


43.508859 | 


42.147193 | 


40.846132 


56 


47.126812 


45.589831 


44.122782 


42.719990 | 


41.381603 


57 


47.827886 


46.243014 


44.731389 


43.287118 | 


41.910128 


58 


48.524605 | 46.891325 


45.334717 


43.848634 | 


42.432773 


59 


49.216997 I 47.534830 


45.932811 | 


44.404588 | 


42.949604 


60 


49.905188 | 


48.173535 | 


46.525715 | 


44.955038 | 


43.460685 



ANNUITIES CERTAIN. 



S9 



TABLE NO. V. Continued. 
The Present Value of One Dollar Payable at the End of 
Each Period at the Rates and for the Terms Stated. au\ 



Periods 


1 1-4 % 


1 3-8 % 


1 1-2 % 


1 5-8 c /o 


1 3-4 % 


1 


.987654 


.986437 


.985222 


.984010 


.982801 


2 


1.963117 


1.959494 


1.955883 


1.952284 


1.948699 


3 


2.926536 


2.91935;? 


2.912200 


2.905076 


2.897984 


4 


3.878061 


3.866193 


3.854385 


3.842633 


3.830943 


5 


4.817838 


4.800191 


4.782645 


4.765198 


4.747855 


6 


5.746613 


5.721521 


5.697187 


5.673011 


5.648998 


7 


6.663329 


6.630354 


6.598214 


6.566308 


6.534641 


8 


7.568828 


7.526861 


7.485925 


7.445321 


7.405053 


9 


8.463049 


8.411208 


8.360517 


8.310278 


8.260494 


10 


9-.346231 


9.283560 


9.222185 


9.161404 


9.101223 


11 


10.218509 


10.144080 


10.071118 


9.998921 


9.927492 


12 


11.080018 


10.992929 


10.907505 


10.823045 


10.739550 


13 


11.930891 


11.830265 


11.731532 


11.633991 


11.537641 


14 


12.771260 


12.656244 


12.543381 


12.431970 


12.322006 


15 


13.601254 


13.471020 


13.343233 


13.217189 


13.092880 


16 


14.421001 


14.274746 


14.131264 


13.989853 


13.850497 


17 


15.230628 


15.067569 


14.907649 


14.750161 


14.595083 


18 


16.030260 


15.849639 


15.672561 


15.498312 


15.326863 


19 


16.820020 


16.621102 


16.426168 


16.234601 


16.046057 


20 


17.600120 


17.382101 


17.168639 


16.959017 


16.752881 


21 


18.370500 


18.132778 


17.900137 


17.671849 


17.447549 


22 


19.131369 


18.873273 


18.620824 


18.373283 


18.130269 


23 


19.882845. 


19.603725 


19.330861 


19.063499 


18.801248 


24 


20.625043 


20.324270 


20.030405 


19.742680 


19.460686 


25 


21.358077 


21.035041 


20.719611 


20.411001 


20.108782 


26 


22.082063 


21.736172 


21.398632 


21.068635 


20.745732 


27 


22.797111 


22.427793 


22.067617 


21.715753 


21.371726 


28 


23.503331 


23.110034 


22.726717 


22.352524 


21.986954 


29 


24.200832 


23.783021 


23.376076 


22.979113 


22.591602 


30 


24.889722 


24.446780 


24.015838 


23.595682 


23.185849 


31 


25.570108 


25.101635 


24.646146 


24.202392 


23.769877 


32 


26.242094 


25.747608 


25.267139 


24.799401 


24.343858 


33 


26.905783 


26.384820 


25.878954 


25.386864 


24.907970 


34 


27.561279 


27.012389 


26.481728 


25.964933 


25.462377 


35 


28.208682 


27.632432 


27.075595 


26.533758 


26.007251 


36 


28.848093 


28.244065 


27.660684 


27.093488 


26.542753 


37 


29.479664 


28.8474U3 


28.237127 


27.644268 


27.069045 


38 


30.103324 


29.442557 


28.805052 


28.186240 


27.586285 


39 


30.719344 


30.029639 


29.364583 


28.719546 


28.094629 


40 


31.327759 


30.608753 


29.915845 


29.244324 


28.594229 


41 


31.928663 


31.180018 


30.458961 


29.760711 


29.085238 


42 


32.522135 


31.743534 


30.994050 


30.268841 


29.567801 


43 


33.108293 


32.299407 


31.521232 


30.769999 


30.042065 


44 


33.687215 


32.847740 


32.040622 


31.262008 


30.508172 


45 


34.258990 


33.38S636 


32.552337 


31.746150 


30.966262 


46 


34.823706 


33.922196 


33.056490 


32.222551 


31.416474 


47 


35.381450 


34.448519 


33.553192 


32.691434 


31.858943 


48 


35.932209 


34.967703 


34.042554 


33.152721 


32.293801 


49 


36.476267 


35.479846 


34.524683 


33.606632 


32.721181 


50 


37.013608 


35.985042 


34.999688 


34.053286 


33.141209 


51 


37.544315 


36.483386 


35.467650 


34.492797 


33.554015 


52 


38.068470 


36.974971 


35.928716 


34.925280 


33.959711 


53 


38.586154 


37.459888 


36.382968 


35.350847 


34.358446 


54 


39.097447 


37.938228 


36.830506 


35.769610 


34.750316 


55 


39.602427 


38.410080 


37.271431 


36.181676 


35.135446 


56 


40.101175 


38.875532 


37.705850 


36.587152 


35.513952 


57 


40.593764 


39.334671 


38.133839 


36.986145 


35.885949 


58 


41.080272 


39.787687 


38.555503 


37.378759 


36.251547 


59 


41.560774 


40.234456 


38.970933 


37.765115 


36.610856 


60 


42.035343 


40.675164 


39.380226 


38.145274 


36.963986 



00 



FINANCE AND LIFE INSURANCE. 



TABLE NO. V. Continued. 
The Present Value of One Dollar Payable at the End of 
Each Period at the Rates and for the Terms Stated, an 



Periods 


1 7-8 % 


2% 


2 1-4 % 


2 1-2 % 


2 3-4 % 


1 


.981595 


.980392 


.977995 


.975610 


.973236 


2 


1.945124 


1.941561 


1.934469 


1.927424 


1.920424 


3 


2.890979 


2.883883 


2.869897 


2.856024 


2.842262 


4 


3.819306 


3.807720 


3.784740 


3.761974 


3.739428 


5 


4.730606 


4.713460 


4.679453 


4.645829 


4.612582 


6 


5.625134 


5.601431 


5.554477 


5.508125 


5.462367 


7 


6.503198 


6.471991 


6.410246 


6.349891 


6.289408 


8 


7.365101 


7.325481 


7.247185 


7.170137 


7.094314 


9 


8.211141 


8.162237 


8.065706 


7.970866 


7.877678 


10 


9.041609 


8.982585 


8.866216 


8.752064 


8.640076 


11 


9.856792 


9.786848 


9.649111 


9.514209 


9.382069 


12 


10.656972 


10.575341 


10.414778 


10.257765 


10.104204 


13 


11.442425 


11.348374 


11.163598 


10.983185 


10.807011 


14 


12.213423 


12.106249 


11.895939 


11.690912 


11.491008 


15 


12.970229 


12.849264 


12.612166 


12.381378 


12.156699 


16 


13.713106 


13.577709 


13.312631 


13.055003 


12.804573 


17 


14.442310 


14.291872 


13.997683 


13.712198 


13.435108 


18 


15.158093 


14.992031 


14.667661 


14.353364 


14.048767 


19 


15.860702 


15.678462 


15.322896 


14.978891 


14.646002 


20 


16.550380 


16.351433 


15.963712 


15.589162 


15.227252 


21 


17.227364 


17.011209 


16.590428 


16.184549 


15.792946 


22 


17.891978 


17.658048 


17.203352 


16.765413 


16.343500 


23 


18.544171 


18.292204 


17.802789 


17.332110 


16.879319 


24 


19.184459 


18.913926 


18.389036 


17.884986 


17.400797 


25 


19.812962 


19.523456 


18.962383 


18.424376 


17.908318 


26 


20.429898 


20.121036 


19.523113 


18.950611 


18.402255 


27 


21.035479 


20.706898 


20.071504 


19.464011 


18.882974 


28 


21.629904 


21.281272 


20.607828 


19.964889 


19.350826 


29 


22.213398 


21.844385 


21.132350 


20.453550 


19.806157 


30 


22.786154 


22.396456 


21.645330 


20.930293 


20.249301 


31 


23.348369 


22.937701 


22.147022 


21.395407 


20.680585 


32 


23.900235 


23.468335 


22.637674 


21.849178 


21.100326 


33 


24.441943 


23.988564 


23.117530 


22.291881 


21.508833 


34 


24.973682 


24.498592 


23.586826 


22.723786 


21.906407 


35 


25.495634 


24.998619 


24.045796 


23.145157 


22.293340 


36 


26.007979 


25.488842 


24.494666 


23.556251 


22.669918 


37 


26.510895 


25.969453 


24.933658 


23.957318 


23.030416 


38 


27.004551 


26.440641 


25.362991 


24.348603 


[ 23.393106 


39 


27.489125 


26.902589 


25.782876 


24.730344 


23.740249 


40 


27.964780 


27.355479 


26.193522 


25.102775 


24.078101 


41 


28.431680 


27.799489 


26.595131 


25.466122 


24.406911 


42 


28.889987 


28.234794 


26.987904 


25.820607 


24.726921 


43 


29.339859 


28.661562 


27.372033 


26.166446 


| 25.038366 


44 


29.781451 


29.079963 


27.747710 


26.503849 


| 25.341475 


45 


30.214916 


29.490160 


28.115120 


26.833024 


f 25.636472 


46 


30.640403 


29.892314 


28.474445 


27.154170 


25.923574 


47 


31.058058 


30.286582 


28.825863 


27.467483 


| 26.202992 


48 


31.468027 


30.673119 


29.169548 


27.773154 


26.474931 


49 


31.870450 


31.052078 


29.505670 


28.071369 


| 26.739592 


50 


32.265466 


31.423606 


29.834396 


28.362312 


| 26.997169 


51 


32.653212 


31.787849 


30.155888 


28.646158 


| 27.247853 


52 


33.033822 


32.144950 


30.470306 


28.923081 


| 27.491827 


53 


33.407426 


32.495049 


30.777805 


29.193250 


| 27.729272 


54 


33.774154 


32.838283 


31.078538 


29.456829 


27.960362 


55 


34.134132 


33.174787 


31.372653 


29.713980 


| 28.185267 


56 


34.487485 


33.504693 


31.660296 


29.964858 


28.404153 


57 


34.834334 


33.828130 


31.941609 


30.209618 


| 28.617180 


58 


35.174800 


34.145225 


32.216733 


30.448407 


| 28.824506 


59 


35.508999 


34.456103 


32.485803 


30.681373 


| 29.026283 


60 


35.837047 


34.760885 


32.748952 


30.908657 


| 29.222660 



ANNUITIES CERTAIN. 



91 



TABLE NO. V. Continued. 
The Present Value of One Dollar Payable at the End of 
Each Period at the Rates and for the Terms Stated, an 



Periods 


3% 


3 1-2 % 


4% 


4 1-2 % 


5% 


1 


.970874 


.966184 


.961538 


.956938 


.95238 


2 


1.913470 


1.899694 


1.886095 


1.872668 


1.85941 


3 


2.828611 


2.801637 | 


2.775091 


2.748964 


2.72325 


4 


3.717098 


3.673079 


3.629895 


3.587526 


3.54595 


5 


4.579707 


4.515052 


4.451822 


4.389977' 


4.32948 


6 


5.417191 


5.328553 


5.242137 


5.157872 


5.07569 


7 


6.230283 


6.114544 


6.002055 


5.892701 


5.78637 


8 


7.019692 


6.873956 


6.732745 


6.595886 


6.46321 


9 


7.786109 


7.607687 


7.435332 


7.268791 


7.10782 


10 


8.530203 


8.316605 


8.110896 


7.912718 


7.72174 


11 


9.252624 


9.001551 


8.760477 


8.528917 


8.30641 


12 


9.954004 


9.663334 


9.385074 


9.118581 


8.86325 


13 


10.634955 


10.302738 


9.985648 


9.682852 


9.39357 


14 


11.296073 


10.920520 


10.563123 


10.222825 


9.89864 


15 


11.937935 


11.517411 


11.118387 


10.739546 


10.37966 


16 


12.561102 


12.094117 


11.652296 


11.234015 


10.83777 


17 


13.166118 


12.651321 


12.165669 


11.707191 


11.27407 ■ 


18 


13.753513 


13.189682 


12.659297 


12.159992 


11.68959 


19 


14.323799 


13.709837 


13.133930 


12.593294 


12.08532 


20 


14.877475 
15.415024 


14.212403 


13.590326 


13.007936 


12.46221 


21 


14.697974 


14.029160 


13.404724 


12.82115 


22 


15.936917 


15.167125 


14.451115 


13.784425 


13.16300 


23 


16.443608 


15.620410 


14.856842 


14.147775 


13.48857 


24 


16.935542 


16.058368 


15.246963 


14.495478 


13.79864 


25 


17.413148 


16.481515 


15.622080 


14.828209 


14.09395 


26 


17.876842 


16.890352 


15.982769 


15.146611 


L 14.37519 


27 


18.327031 


17.285365 


16.329586 


15.451303 


14.64303 


28 


18.764108 


17.667019 


16.663063 


15.742873 


14.89813 


29 


19.188455 


18.035767 


16.983715 


16.021889 


15.14107 


30 


19.600441 


18.392045 


17.292033 


16.288888 


15.37245 


31 


20.000428 


18.736276 


17.588494 


16.544391 


15.59281 


32 


20.388766 


19.068865 


17.873552 


16.788891 


15.80268 


33 


20.765792 


19.390208 


18.147646 


17.022862 


16.00255 


34 


21.131837 


19.700684 


18.411198 


17.246758 


16.19290 


35 


21.487220 


20.000661 


18.664613 


17.461012 


16.37419 


36 


21.832253 


20.290494 


18.908282 


17.666041 


16.54685 


37 


22.167235 


20.570525 


19.142579 


17.862239 


16.71129 


38 


22.492461 


20.841087 


19.367864 


18.049990 


16.86789 


39 


22.808215 


21.102500 


19.584485 


18.229656 


17.01704 


40 


23.114772 


21.355072 


19.792774 


18.401584 


17.15909 


41 


23.412400 


21.599104 


19.993052 


18.566109 


17.29437 


42 


23.701359 


21.834883 


20.185627 


18.723550 


17.42321 


43 


23.981902 


22.062689 


20.370795 


18.874210 


17.54591 


44 


24.254274 


22.282791 


20.548841 


19.018383 


17.66277 


45 


24.518713 


22.495450 


20.720040 


19.156347 


17.77407 


46 


24.775449 


22.700918 


20.884654 


19.288370 


17.88007 


47 


25.024708 


22.899438 


21.042936 


19.414709 


17.98102 


48 


25.266707 


23.091214 


21.195131 


19.535607 


• 18.07716 


49 


25.501657 


23.276565 


21.341472 


19.651298 


18.16872 


50 


25.729764 


23.455618 


21.482185 


19.962008 


18.25593 


51 


25.951227 


23.628616 


21.617485 


19.867949 


18.33898 


52 


26.166240 


23.795764 


21.747582 


19.969329 


18.41807 


53 


26.374990 


23.957260 


21.872675 


20.066344 


18.49340 


54 


26.577660 


24.113395 


21.992957 


20.159180 


18.56515 


55 


26.774427 


24.264053 


22.108612 


20.248019 


18.63347 


56 


26.965463 


24.409713 


22.219819 


20.333032 


18.69854 


57 


27.150935 


24.550447 


22.326749 


20.414385 


18.76052 


58 


27.331005 


24.686422 


22.429566 


20.492234 


18.81954 


59 


27.505830 


24.817899 


22.528429 


20.566731 


18.87575 


60 


27.675563 


24.944733 


22.623489 


20.638020 


18.92929 



92 



FINANCE AND LIFE INSURANCE. 



TABLE NO. V. Concluded. 
The Present Value of One Dollar Payable at the End of 
Each Period at the Rates and for the Terms Stated, am 



Periods 


6% 


7% 


8% 


9% 


10% 


1 | 


.94339 


.93458 | 


.92593 


.91743 | 


.90909 


2 I 


1.83340 


1.80802 j 


1.78327 


1.75911 | 


1.73554 


3 I 


2.67301 


2.62432 1 


2.57710 


2.53129 | 


2.48685 


4 I 


3.46511 


3.38721 


3.31213 


3.23972 J 


3.16987 


5 


4.21236 


4.10020 


3.99271 


3.88965 


3.79079 


6 I 


4.91732 


4.76654 


4.62288 


4.48592 | 


4.35526 


7 1 


5.58238 


5.38929 


5.20637 


5.03295 


4.86842 


8 


6.20979 


5.97130 


5.74664 


5.53482 


5.33493 


9 1 


6.80169 


6.51523 


6.24689 


5.99525 


5.75902 


10 | 


7.36009 


7.02358 


6.71008 


6.41766 


6.14457 


11 1 


7.88687 


7.49867 


7.13896 


6.80519 


6.49506 


12 | 


8.38384 


7.92469 


7.53608 


7.16072 


6.81369 


13 | 


8.85268 


8.35765 


7.90378 


7.48690 


7.10336 


14 


9.29498 


8.74547 


8.24424 


7.78615 


7.36669 


15 


9.71225 


9.10791 


8.55948 


8.06069 


7.60608 


16 


10.10589 


9.44665 


8.85137 


8.31256 


7.82371 


17 


10.47726 


9.76322 


9.12164 


8.54362 


8.02155 


18 | 


10.82760 


10.05909 


9.37189 


8.75562 


8.20141 


19 | 


11.15812 


10.33559 


9.60360 


8.95011 


8.36492 


20 | 


11.46992 


10.59401 


9.81815 


9.12854 


8.51356 


21 | 


11.76408 


10.83553 


10.01680 


9.29224 


8.64869 


22 | 


12.04158 


11.06124 


10.20074 


9.44242 


8.77154 


23 | 


12.30338 


11.27219 


10.37106 


9.58021 


8.88322 


24 


12.55036 


11.46933 


10.52876 


9.70661 


8.98474 


25 ] 


12.78336 


11.65358 


10.67478 


9.82258 


9.07704 


26 | 


13.00317 


11.82578 


10.80998 


9.92897 


9.16095 


27 | 


13.21053 


11.98671 


10.93552 


10.02658 


9.23722 


28 | 


13.40616 


12.13711 


11.05108 


10.11616 


9.30657 


29 | 


13.59072 


12.27767 


11.15841 


10.19832 


9.36961 


30 | 


13.76483 


12.40904 


11.25728 


s 10.27369 


9.42691 


31 | 


13.92909 


12.53181 


11.34980 


10.34284 


9.47901 


32 


14.08404 


12.64656 


11.43499 


10.40627 


9.52638 


33 


14.23023 


12.75379 


11.51389 


10.46447 


9.56943 


34 | 


14.36814 


12.85401 


11.58694 


10.51787 


9.60858 


35 | 


14.49825 


12.94767 


11.65457 


10.56685 


9.64416 


36 


14.62099 


13.03521 


11.71719 


10.61179 


9.67651 


37 


14.73678 


13.11702 


11.77518 


10.65302 


9.70592 


38 


14.84602 


13.19347 


11.82887 


10.69085 


9.73265 


39 


14.94908 


13.26493 


11.87858 


10.72555 


9.75696 


40 


15.04630 


13.33171 


11.92461 


10.75739 


9.77905 


41 


15.13802 


13.39412 


11.96724 


10.78659 


9.79914 


42 


15.22454 


13.45245 


12.00670 


10.81339 


9.81740 


43 


15.30617. 


13.50696 


12.04324 


10.83797 


9.83399 


44 


15.38318 


13.55791 


12.07707 


10.86053 


9.84909 


45 


15.45583 


13.60552 


12.10840 


10.88122 


9.86281 


46 


15.52437 


13.65002 


12.13741 


10.90020 


9.87528 


47 


15.58903 


13.69161 


12.16427 


10.91762 


9.88662 


48 


15.65003 


13.73047 


12.18914 


10.93360 


9.89693 


49 


15.70757 


13.76680 


12.21216 


10.94826 


9.90630 


50 


15.76186 


13.80075 


12.23348 


10.96170 


9.91481 


51 


15.81308 


13.83247 


12.25323 


10.97404 


9.92256 


52 


15.86139 


13.86212 


12.27151 


10.98536 


9.92960 


53 


15.90697 


13.88984 


12.28843 


10.99575 


9.93599 


54 


15.94998 


13.91574 


12.30410 


11.00527 


9.94182 


55 


15.99054 


13.93994 


12.31861 


11.01401 


9.94711 


56 


16.02881 


13.96256 


12.33205 


11.02203 


9.95192 


57 


16.06492 


13.98370 


12.34449 


11.02939 


9.95629 


58 


16.09898 


14.00346 


12.35601 


11.03614 


9.96026 


59 


16.13111 


14.02193 


' 12.36667 


11.04233 


9.96387 


60 


| 16.16143 


14.03918 


12.37655 


11.04801 


9.96715 



94 



FINANCE AND LIFE INSURANCE. 



TABLE NO. VI. 

The Annuity Which "Will Amount to One Dollar^ in a 

Stated Term at the Rates of Interest Stated. Snl 



Periods 



1-4% 



1-3 % 3-8 % 



5-12 



1-2% 



1 


1.000000 


1.000000 


1.000000 


1 1.000000 


1.000000 


2 


.499401 


.499102 


.499068 


.498982 


.498753 


3 


.332491 


.332206 


.332094 


.331979 


.331672 


4 


.249053 


.248713 


.248597 


1 .248465 


.248133 


5 


.198997 


.198653 


.198507 


.198353 


.198010 


6 


.165629 


.165270 


.165111 


.164951 


.164595 


7 


.141788 


.141421 


.141259 


.141106 


.140729 


8 


.123910 


.123526 


.123370 


.122913 


.122829 


9 


.110006 


.109772 


.109706 


.109281 


.108907 


10 


.098888 


.098499 


.098326 


.098146 


.097771 


1 


.089780 


.089404 


.089218 


.089031 


.088659 


2 


.082194 


.081816 


.081630 


.081253 


.081066 


3 


.075776 


.075396 


.075208 


.075017 


.074642 


4 


.070276 


.069894 


.069704 


.069514 


.069136 


5 


.065508 


.065125 


.064937 


.064744 


.064364 


6 


.061336 


.060952 


.060761 


.060572 


.060189 


7 


.057655 


.057269 


.057079 


.056887 


.056506 


8 


.054384 


.053998 


.053806 


.053614 


.053232 


9 


.051457 


.051071 


.050878 


.050648 


.050303 


20 


.048822 


.048435 


.048242 


.048049 


.047666 


1 


.046439 


.046051 


.045858 


.045665 


.045282 


2 


.044273 


.043884 


.043691 


.043589 


.043114 


3 


.042295 


.041906 


.041711 


.041519 


.041135 


4 


.040481 


.040092 


.039898 


.039705 


.039321 


5 


.038813 


.038423 


.038229 


.038036 


037652 


6 


.037273 


.036882 


.036678 


.036528 


.036112 


7 


.035847 


.035457 


.035264 


.035071 


.034686 


8 


.034524 


.034133 


.034017 


.033746 


. .033362 


9 


.033291 


.032900 


.032706 


.032511 


.032129 


30 


.032140 


.031970 


.031483 


.031389 


.030978 


1 


.031064 


.030674 


.030480 


.030286 


.029903 


2 


.030056 


.029665 


.029571 


.029278 


.028895 


3 


.029108 


.028717 


.028518 


.028331 


.027947 


4 


.028216 


.027826 


.027695 


.027438 


.027056 


5 


.027375 


.026985 


.026787 


.026589 


.026216 


6 


.026581 


.026191 


.025997 


.025804 


.025422 


7 


.025830 


.025381 


.025246 


.025053 


.024671 


8 


.025118 


.024928 


.024535 


.024342 


.023960 


9 


.024443 


.024053 


.023860 


.023667 


.023286 


40 


.023802 


.023412 


.023272 


.023026 


.022646 


1 


.023197 


.022750 


.022609 


.022417 


.022036 


2 


.022611 


.022221 


.022028 


.021836 


.021456 


3 


.022057 


.021668 


.021475 


.021283 


.020903 


4 


.021529 


.021139 


.020946 


.020851 


.020375 


5 


.021023 


.020634 


.020488 


.020250 


.019871 


6 


.020540 


.020151 


.019959 


.019767 


.019389 


7 


.020077 


.019689 


.019495 


.019305 


.018927 


8 


.019634 


.019246 


.019053 


.018863 


.018485 


9 


.019209 


.018821 


.018629 


.018438 


.018061 


50 


.018801 


.018413 


.018221 


.018030 


.017654 


1 


.018395 


.018021 


.017788 


.017639 


.017261 


2 


.018032 


.017644 


.017453 


.017262 


.016886 


3 


.017669 


.017282 


.017090 


.016901 


.016525 


4 


.017320 


.016933 


.016741 


.016552 


.016214 


5 


.016983 


.016596 


.016405 


.016216 


.015841 


6 


.016659 


.016272 


.016081 


.015892 


.015516 


7 


.016345 


.015959 


.015768 


.015579 


.015206 


8 


.016043 


'.015657 


.015466 


.015278 


.014905 


9 


.015751 


.015365 


.015175 


.014986 


.014614 


60 


.015469 


.015083 


.014893 


.014705 


.014333 



SINKING FUNDS. 

TABLE NO. VI. Continued. 

The Annuity Which Will Amount to One Dollar in a 

Stated Term at the Rates of Interest Stated. Sni 



Periods 


1-4% 


1-3% 


3-8 % 


5-12 % 


1-2% 


61 


.015196 


.014810 


.014622 


.014432 


.014061 


2 


.014932 


.014546 


.014357 


.014169 


.013798 


3 


.014676 


.014291 


.013974 


.013913 


.013543 


4 


.014428 


.014043 


.013854 ' 


.013666 


.013297 


5 


.014187 


.013804 


.013614 


.013427 


.013058 


6 


.013955 


.013571 


.013382 


.013195 


.012826 


7 


.013729 


.013335 


.013157 


.012971 


.012602 


8 


.013509 


.013126 


.012937 


.012751 


.012384 


9 


.013297 


.012914 


.012726 


.012539 


.012172 


70 


.013090 


.012708 


.012519 


.012333 


.011967 


1 


.012887 


.012507 


.012319 


.012133 


.011767 


2 


.012694 


.012312 


.012124 


.011938 


.011576 


3 


.012504 


.012122 


.011935 


.011749 


.011384 


4 


.012319 


.011938 


.011750 


.011565 


.011201 


5 


.012139 


.011758 


.011598 


.011386 


.011022 


6 


.011965 


.011583 


.011396 


.011212 


.010848 


7 


.011793 


.011413 


.011226 


.011042 


.010679 


8 


.011627 


.011245 


.011086 


.010876 


.010514 


9 


.011465 


.011086 


.010900 


.010715 


.010354 


80 


.011307 


.010928 


.010742 


.010558 


.010197 


1 


.011153 


.010774 


.010579 


.010404 


.010044 


2 


.011008 


.010624 


.010439 


.010255 


.009896 


3 


.010856 


.010478 


.010293 


.010110 


.009750 


4 


.010713 


.010335 


.010150 


.009967 


.009609 


5 


.010574 


.010196 


.010104 


.009828 


.009470 


6 


.010437 


.010060 


.009875 


.009693 


.009335 


7 


.010314 


>009927 


.009742 


.009560 


.009203 


8 


.010173 


.009797 


.009613 


.009431 


.009074 


9 


.010046 


.009670 


.009486 


.009304 


.008948 


90 


.009922 


.009546 


.009362 


.009181 


.008825 


1 


.009800 


.009425 


.009241 


1 .009060 


.008705 


2 


.009681 


.009306 


.009122 


.008941 


.008587 


3 


.009564 


.009190 


.008998 


.008826 


1 .008472 


4 


.009494 


.009076 


.008889 


.008713 


.008060 


5 


.009339 


.008965 


.008782 


1 .008602 


1 .008249 


6 


.009230 


.008856 


.008673 


I .008493 


.008441 


7 


.009123 


.008749 


.008567 


1 .008387 


1 .008036 


8 


.009017 


.008645 


.008463 


.008287 


.007902 


9 


.008915 


.008540 


1 .008361 


I .008180 


.0D7831 


100 


.008815 


| .008442 


! .008260 


| .008081 


1 .007732 


1 


I .008716 


.008344 


.008162 


.007984 


1 .00763~5 


2 


.008619 


.008248 


I .008066 


! .007887 


1 .007539 


3 


.008524 


.008153 


1 .007972 


.007794 


I .007446 


4 


.008431 


.008061 


.007880 


.007702 


.007355 


5 


1 .008340 


.007970 


.007789 


1 .007612 


! .007265 


6 


I .008251 


1 .007881 


1 .007700 


1 .007524 


I .007177 


7 


.008163 


.007793 


1 .007613 


! .007436 


1 .007090 


8 


.008077 


.007709 


1 .007528 


! .007351 


I .007006 


9 


.007992 


1 .007624 


I .007444 


1 .007267 


i .006922 


110 


! .007909 


1 .007541 


I .007361 


! .007185 
I .007104 


I .006841 


1 


.007828 


.007461 


I .007280 


1 .006761 


2 


.007748 


.007382 


I .007201 


I .007025 


1 .006682 


3 


.007G69 


.007302 


.007123 


I .006946 


1 .006605 


4 


.007592 


.007225 


.007046 


1 .00P871 


! .006529 


5 


! .007516 


1 • .007150 


.006971 


1 .006796 


! .006455 


6 


1 .007442 


1 .007075 


I .006897 


1 .006722 


I .006382 


7 


1 .007369 


.007002 


1 .006824 


! .006647 


! .006310 


8 


.007297 


1 .006931 


.006753 


I .006579 


.006240 


9 


.007226 


.006861 


! .006682 


I .006509 


1 .006170 


120 


| .007156 


! .006791 


I .006614 


J .006444 


1 .006102 



96 



FINANCE AND LIFE INSURANCE. 



TABLE NO. VI. Continued. 

The Annuity Which Will Amount to One Dollar in a 
Stated Term at the Rates of Interest Stated. Snl" 1 



Periods 


5-8% 


3-4% 


7-8% 


1% 


1 1-8 % 


1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


.498524 


.498140 


.497822 


.497512 


.497202 


3 


.331266 


.330848 


.330434 


.330022 


.329612 


4 


.247672 


'.247207 


.246743 


.246281 


.245821 


5 


.197516 


.197021 


.196530 


.196040 


.195555 


6 


.164081 


.163570 


.163138 


.162548 


.162041 


7 


.140201 


.139672 


.139151 


.138628 


.138108 


8 


.122290 


.121755 


.121222 


.120690 


.120162 


9 


.108362 


.107819 


.107279 


.106740 


.106205 


10 


.097220 


.096671 


.096125 


.095582 


.095056 


1 


.088104 


.087550 


.087001 


.086454 


.085910 


2 


.080507 


.079910 


.079399 


.078849 


.078302 


3 


.074080 


.073523 


.072967 


.072415 


.071867 


4 


. .068572 


.068012 


.067455 


.066901 


.066362 


5 


.063798 


.063210 


.062678 


.062124 


.061573 


6 


.059623 


.059059 


.058500 


.057945 


.057394 


7 


.055937 


.059373 


.054813 


.054258 


.053708 


8 


.052706 


.052098 


.051538 


.050982 


.050431 


9 


.049733 


.049164 


.048607 


.048052 


.047501 


20 


.046988 


.046531 
.044120 


.045970 


.045415 


.047501 


1 


.044711 


.043585 


.043031 


.042480 


2 


.042543 


.041978 


.041418 


.040864 


.040316 


3 


.040564 


.039985 


.039439 


.038886 


.038338 


4 


.038774 


.038185 


.037626 


.037073 


.036527 


5 


.037081 


.036517 


.035958 


.035407 


.034862 


6 


.035521 


.034977 


.034420 


.033869 


.033325 


7 


.034115 


.033556 


.032995 


.032446 


.031903 


8 


.032867 


.032229 


.031673 


.031124 


.030583 


9 


.031560 


.030998 


.030442 


.029895 


.029355 


30 


.030411 


.029848 


.029294 


.028748 


.028210 


1 


.029334 


.028774 


.028221 


.027676 1 


.027139 


2 


.028326 


.027765 


.027215 


.026671 ! 


.026135 


3 


.027380 


.026822 


.026270 


.025727 ! 


.025194 


4 


.026489 


.025931 


.025381 1 


.024840 1 


.024308 


5 


.025649 


.025092 


.024543 


.024004 


.023473 


6 


.024856 1 


.024300 


.023752 


.023214 1 


.022685 


7 


.024106 


.023551 


.023005 


.022468 1 


.021941 


8 1 


.023396 . 


|022842 1 


.022297 1 


.021761 1 


.021236 


9 ! 


.022723 1 


.022169 


.021625 1 


.021092 I 


.020568 


40 | 


.022083 | 


.021531 1 


.020988 1 


.020455 ! 


.019934 


1 1 


.021414 1 


.020924 1 


.020382 I 


.019851 ! 


.019332 


2 1 


.020875 1 


.020345 1 


.019805 1 


.019276 1 


.018757 


3 1 


.020343 1 


.019794 1 


.019255 1 


.018727 1 


.018211 


4 ! 


.019816 ! 


.019267 1 


.018730 1 


.018204 I 


.017690 


5 1 


.019312 1 


.018765 1 


.018229 1 


.017795 1 


.017192 


6 1 


.018831 1 


.018285 1 


.017751 ! 


.017228 I 


.016719 


7 1 


.018371 I 


.017825 1 


.017292 ! 


.016771 ! 


.016262 


8 1 


.017929 1 


.017383 I 


.016853 ! 


,016333 1 


.015826 


9 ! 


.017506 1 


.016983 1 


.016433 I 


.015915 ! 


.015409 


50 I 


.017099 I 


.016558 1 


.016029 I 


.015513 I 


.015009 


1 I 


.016708 I 


.016189 I 


.015641. I 


.015127 1 


.014629 


2 I 


.016334 1 


.015795 1 


.015269 1 


.014756 ! 


.014256 


3 


.015973 I 


.015436 I 


.014911 1 


.014400 1 


.013902 


4 ! 


.015626 1 


.015089 1 


.014566 I 


.014957 I 


.013560 


5 


.015292 


.014756 


.014234 I 


.013726 1 


.013232 


6 


.014969 1 


.014435 I 


.013994 1 


.013408 1 


.012917 


7 


.014658 


.014125 


.013606 1 


.013102 1 


.01.2611 


8 


.014358 


'.013826 


.013309 ! 


.012806 1 


.012317 


9 


.014067 


.013537 


.013021 


.012520 ! 


.012164 


60 


.013786 


.013258 


.012744 | 


.012244 | 


.011760 



SINKING FUNDS. 



97 



TABLE NO. VI. Continued. 



The Annuity Which Will Amount to One Dollar_ in 
Stated Term at the Rates of Interest Stated. Snl 



Periods 


1 1-4 % 


1 3-8 % 


1 1-2 % 


1 5-8 % 


1 3-4 % 


1 


1.000000 


1.000000 


1.00000C 


1.000000 


1.000000 


2 


.496900 


.496587 


.496278 


.495971 


.495663 


3 


.329200 


.328450 


.328383 


.327971 


.327567 


4 


.245356 


.244897 


.244445 


.243990 


.243532 


5 


.195062 


.194569 


.194089 


.193604 


.193121 


6 


.161534 


.161184 


.160525 


.160022 


.159523 


7 


.137437 


.137071 


.136556 


.136045 


.135531 


8 


.119909 


.119107 


.118584 


.118062 


.117543 


9 


.105760 


.105139 


.104610 


.104087 


.103558 


10 


.094285 


.093984 


.093434 


.092904 


.092375 


1 


.085369 


.084830 


.084294 


.083761 


.083230 


2 


.077759 


.077218 


.076680 


.076146 


.075614 


3 


.071321 


.070779 


.070240 


.069705 


.069173 


4 


.065805 


.065262 


.064723 


.064188 


.063656 


5 


.061026 


.060483 


.059944 


.059409 


.058877 


6 


.056844 


.056307 


.055765 


.055230 


.054699 


7 


.053160 


.052619 


.052080 


.051546 


.051016 


8 


.049885 


.049343 


.048806 


.048273 


.047745 


9 


.046956 


.046415 


.045878 


.045347 


.044821 


20 


.044320 


.043780 


.043246 


.042716 


.042191 


1 


.041938 


.041399 


.040865 


.040337 


.039815 


2 


.039393 


.039235 


.038703 


.038177 


.037656 


3 


.037797 


.037251 


.036731 


.036206 


.035688 


4 


.035936 


.035452 


.034924 


.034403 


.033886 


5 


.034243 


.033790 


.033263 


.032744 


.032230 


6 


.032787 


.032256 


.031732 


.031214 


.030703 


7 


.031295 


.030838 


.030315 


.029800 


.029291 


8 


.030049 


.029521 


.029001 


.028488 


.027982 


9 


.028822 


.028297 


.027779 


.027268 


.026764 


30 


.027679 
.026616 


.027155 


.026639 


.026131 


.025630 


1 


.026088 


.025574 


.025068 


.024570 


2 


.025608 


.025088 


.024577 


.024070 


.023578 


3 


.024668 


.024151 


.023641 


.023141 


.022648 


4 


.023784 


.023267 


.022762 


.022264 


.021774 


5 


.022951 


.022438 


.021934 


.021438 


.020951 


6 


.022165 


.021654 


.021152 


.020659 


.020175 


7 


.021373 


.020914 


.020414 


.019878 


.019443 


8 


.202720 


.020213 


.019716 


.019228 


.018750 


9 


.020054 


.019549 


.019055 


.018570 


.018094 


40 


.019420 


1 .018919 


.018427 


.017945 


1 .017472 


1 


.018821 


.018321 


.017813 


.017351 


.016882 


2 


.018249 


.017751 


.017264 


.016787 


.016321 


3 


.017705 


.017209 


.016725 


.016250 


.015787 


4 


.017146 


.016694 


.016210 


.015739 


.015278 


5 


.016990 


I .016199 


.015720 


I .015251 


.014793 


6 


.016217 


.015728 


.015251 


.014785 


.014330 


7 


.015764 


.015278 


.014803 


.014340 


.013888 


8 


.415331 


1 .014830 


! .014375 


! .013915 


.013466 


9 


.014916 


.014434 


.013965 


.013507 


.013061 


50 


.014518 


I .014039 


.013571 


1 .013117 


I .012674 


1 


.014136 


.013659 


.013105 


1 .012743 


.012303 


2 


.013769 


.013295 


.012833 


.012384 


.011947 


3 


.01.3417 


.012945 


1 .012485 


I .012039 


1 .011605 


4 


.013078 


I .012608 


.012151 


I .011708 


! .011277 


5 


1 .012747 


.012284 


1 .011830 


I .011392 


.010961 


6 


.012436 


.011964 


.011521 


.011082 


! .010658 


7 


.012135 


.011672 


! .011224 


! .010788 


I .010366 


8 


.011845 


.011363 


.010937 


1 .010504 


1 .010085 


9 


.011563 


.011104 


' .010660 


1 .010230 


1 .009814 


60 


.011289 


! .010834 


! .010394 


I .009967 


1 .009553 



98 FINANCE AND LIFE INSURANCE. 

TABLE NO. VI. Continued, 

The Annuity Which Will Amount to One Dollar^ in a 
Stated Term at the Rates of Interest Stated. Sill 1 



Periods 


1 7-8 % 


2% 


2 1-4 % 


2 1-2 % 


2 3-4 % 


1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


.495363 


.495050 


.494438 


.493827 


.493218 


3 


.327162 


.326755 


.325945 


.325137 


.324332 


4 


.243077 


.242624 


.241719 


.240818 


.239921 


5 


.192639 


.192158 


.191200 


.190247 


.189298 


6 


.158932 


.158526 


.157535 


.156550 


.155571 


7 


.135021 


.134512 


.133500 


.132495 


.131497 


8 


.117025 


.116510 


.115485 


.114467 


.113458 


9 


.103149 


.102515 


.101482 


.100457 


.099441 


10 


.091850 


.091327 


.090288 


.089259 


.088240 


1 


.082703 


.082178 


.081136 


.080106 


.079086 


2 


.075086 


.074560 


.073517 


.072487 


.071469 


3 


.068644 


.068118 


.067077 


.066048 


.065033 


4 


.063021 


.062602 


.061562 


.060537 


.059525 


5 


.058350 


.057825 


.056789 


.055766 


.054757 


6 


.054173 


.053650 


.052617 


.051599 


.050597 


7 


.050491 


.049970 


.048940 


.047928 


.046932 


8 


.047191 


.046702 


.045677 


.044670 


.043681 


9 


.044299 


.043782 


.042762 


.041761 


.040778 


20 


.041672 


.041157 


.040142 


.039147 


.038171 


1 


.039297 


.038785 


.037776 


.036787 


.035819 


2 


.037141 


.036631 


.035628 


.034647 


.033686 


3 


.035175 


.034668 


.033671 


.032696 


.031744 


4 


.033375 


.032871 


.031880 


.030913 


.029969 


5 


.031722 


.031220 


.030236 


.029276 


.028340 


6 


.030198 


.029699 


.028721 


.027769 


.026841 


7 


.028789 


.028293 


.027322 


.026377 


.025458 


. 8 


.027482 


.026990 


.026025 


.025088 


.024177 


9 


.026268 


.025778 


.024821 


.023891 


.022989 


30 


.025136 


.024650 


.023699 


.022778 


.021884 


1 


.024024 


.023596 


.022653 


.021739 


.020854 


2 


.023091 


.022611 


.021674 


.020768 


.019892 


3 


.022163 


.021687 


.020757 


.019859 


.018993 


4 


.021290 


.020819 


.019897 


.019007 


.018149 


5 


.020472 


.020002 


.019087 


.018206 


.017356 


6 


.019709 


.019233 


.018325 


.017452 


.016611 


7 


.018970 


.018507 


.017606 


.016741 


.015910 


8 


.018461 


.017821 


.016927 


.016070 


.015248 


9 


.017628 


.017171 


.016285 


.015436 


.014623 


40 


.017009 


.016556 


.015677 


.014836 


.014032 


1 


.016322 


.015972 


.015101 


.014268 


.013472 


2 


.015864 


.015417 


.014554 


.013729 


.012942 


3 


.015690 


.014890 


.014034 


.013217 


.012439 


4 


.014828 


.014388 


.013539 


.012730 


.011961 


5 


.014346 


.013910 


.013068 


.012268 


.011507 


6 


.013887 


.013453 


.012619 


.011827 


.011075 


7 


.013448 


.013018 


.012191 


.011407 


.010664 


8 


.013027 


.012602 


.011782 


.011101 


.010272 


9 


.012627 


.012204 


.011392 


.010623 


.009898 


50 


.012243 


.011823 


.011018 


.010258 


.009541 


1 


.011875 


.011459 


.010661 


.009909 


.009200 


2 


.011522 


.011109 


.010319 


.009575 


.008874 


3 


.011183 


.010774 


.009991 


.009255 


.008563 


4 


.010858 


.010452 


.009676 


.008952 


.008265 


5 


.010544 


.010143 


.009375 


.008654 


.007980 


6 


.010246 


.009884 


.009082 


.008372 


.007706 


7 


.009957 


.009561 


.008807 


.008102 


.007443 


8 


.009679 


.009287 


.008540 


.007843 


.007193 


9 


.009412 


.009022 


.008283 


.007593 


.006952 


60 


.009154 


.008768 


.008035 


.007353 


.006720 



SINKING FUNDS. 



99 



TABLE XO. VI. Continued. 



The Annuity Which Will Amount to One Dollar^ in a 
Stated Term at the Rates of Interest Stated. Sn I 



Periods 


Z% 


3 1-2 % 


4% 


4 1-2 % 


5% 


1 1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


.492611 


.491400 


.490196 


.488998 


.487805 


3 


.323530 


.321934 


.320348 


.318733 


.317209 


4 


.239027 


.237251 


.235490 


.233744 


.232012 


5 


.188355 


.186481 


.184627 


.182792 


.180975 


6 


.154598 


.152668 


.150762 


.148078 


.147017 


7 


.130506 


.128544 


.126610 


.124701 


.122820 


8 


.112456 


.110477 


.108528 


.106610 


.104722 


9 


.098434 


.096446 


.094493 


.092574 


.090690 


10 


.087230 


.085241 


.083290 


.081379 


.079505 


1 


.078077 


.076092 


.074149 


.072248 


.070389 


2 


.070462 


.068484 


.066552 


.064666 


.062825 


3 


.064029 


.062062 


.060144 


.058275 


.056456 


4 


.058526 


.056571 


.054669 


.052820 


.051024 


5 


.053767 


.051825 


.049941 


.048114 


.046342 


6 


.049611 


.047685 


.045820 


.044015 


.042270 


7 


.045952 


.044043 


.042198 


.040418 


.038699 


8 


.042709 


.040817 


.038933 


.037237 


.035546 


9 


.039814 


.037940 


.036139 


.034407 


.032745 


20 


.037216 


.035361 


.033582 


.031876 


.030243 


1 


.034872 


.033037 


.031280 


.029601 


.027996 


2 


.032747 


.030932 


.029199 


.027546 


.025970 


3 


.030S14 


.029019 


.027309 


.025682 


.024137 


4 


.029047 


.027273 


.025587 


.023987 


.022471 


5 


.027428 


.025674 


.024012 


.022439 


.020952 


6 


.025938 


.024205 


.022567 


.021021 


.019564 


7 


.024564 


.022852 


.021238 


.019719 


.018292 


8 


.023293 


.021603 


.020013 


.018521 


.017123 


9 


.022115 


.020445 


.018880 


.017415 


.016046 


30 


.021019 


.019371 


.017830 


.016392 


.015051 


1 


.019999 


.018372 


.016855 


.015443 


.014132 


2 


.019046 


.017442 


.015949 


.014563 


.013280 


3 


.018156 


.016572 


.015104 


.013744 


.012490 


4 


.017322 


.015759 


.014315 


.012982 


.011755 


5 


.016539 


.014998 


.013577 


.012270 


.011072 


6 


.015804 


.0142S4 


.012887 


.011606 


.010434 


7 


.015112 


.013613 


.012239 


.010984 


.009840 


8 


.014459 


.012982 


.011632 


.010402 


.009284 


9 


.013844 


.012388 


.011061 


.009856 


.008764 


40 


.013262 


.011827 


.010523 


.009343 


.008278 


1 


.012712 


.011298 


.010017 


.008862 


.007822 


2 


.012192 


.010798 


.009540 


.008409 


.007395 


3 


.011698 


.010325 


.009090 


.007982 


.006993 


4 


.011230 


.009878 


.008664 


.007581 


.006616 


5 


.010785 


.009453 


.008262 


.007202 


.006262 


6 


.010363 


.009051 


.007882 


.006845 


.005928 


7 


.009961 


.008669 


.007522 


.006507 


.005614 


8 


.009578 


.008306 


.007181 


.006189 


.005318 


9 


.009213 


.007962 


.006857 


.005887 


.005040 


50 


.008866 


.007634 


.006550 


.005602 


.004777 


1 


.008534 


.007322 


.006259 


.005332 


.004529 


2 


.008217 


.007024 


.005982 


.005077 


.004294 


3 


.007915 


.008741 


.005719 


.004835 


.004073 


4 


.007626 


.006471 


.005469 


.004605 


.003864 


5 


.007349 


.006213 


.005231 


.004388 


.003667 


6 


.007084 


.005967 


.005005 


.004181 


.003480 


7 


.006831 


.005732 


.004789 


.003985 


.003303 


8 


.006588 


.005508 


.004584 


.003799 


■ .003136 


9 


.006356 


.005294 


.004388 


.003622 


.002978 


60 


.006133 


| .005080 


.004202 


.003454 


.002828 



100 



FINANCE AND LIFE INSURANCE. 



TABLE NO. VI. Concluded. 

The Annuity Which Will Amount to One Dollar in a 
Stated Term at the Rates of Interest Stated. Sn" 1 



Periods 


6% 


7% 


8% 


9% 


10% 


1 1 


1.000000 


1.000000 


1.000000 


1.000000 


1.000000 


2 


.485437 


.483091 


.480769 


.478469 


.476190 


3 


.314110 


.311056 


.308039 


.305055 


.302115 


4 


.228591 


.225226 


.221920 


.218668 


.215471 


5 


.177396 


.173891 


.170456 


.167092 


.163797 


6 


.143363 


.139791 


.136316 


.132917 


.129607 


7 


.119135 


.115554 


.112073 


.108691 


.105406 


8 


.101036 


.097467 


.094015 


.090674 


.087444 


9 


.087022 


.083866 


.080080 


.076799 


.073641 


10 


.075868 


.073278 


.069065 


.065820 


.062745 


1 


.066793 


.063503 


.060076 


.056934 


.053963 


2 


.059277 


.055901 


.052695 


.049652 


.046763 


3 


.052960 


.049765 


.046522 


.043567 


.040779 


4 


.047585 


.044345 


.041297 


.038433 


.035747 


5 


.042963 


.039795 


.036830 


.034059 


.031474 


6 


.038952 


.035858 


.032977 


.030300 


.027817 


7 


.035445 


.032425 


.029617 


.027046 


024664 


8 


.032357 


.029413 


.026702 


.024212 


.021930 


9 


.029621 


.026753 


.024128 


.021730 


.019547 


20 


.027185 


.024393 


.021852 


.019546 


.017460 


1 


.025005 


.022289 


.019832 


.017617 


.015624 


2 


.023046 


.020405 


.018032 


.015905 


.014005 


3 


.021278 


.018714 


.016421 


.014382 


.012572 


4 


.019679 


.017189 


.014978 


.013023 


.011300 


5 


.018227 


.015681 


.013679 


.011806 


.010168 


6 


.016904 


.014561 


.012507 


.010715 


.009159 


7 


.015697 


.013426 


.011448 


.009735 


.008277 


8 


.014593 


.012392 


.010489 


.008852 


.007451 


9 


.013580 


.011449 


.009619 


.008056 


.006728 


30 


.012649 


.010586 


.008827 


.007336 


.006079 


1 


.011792 


.009797 


.008107 


.006686 


.005496 


2 


.011002 


.009073 


.007451 


.006096 


:004972 


3 


.010273 


.008408 


.006852 


.005562 


.004499 


4 


.009598 


.007797 


.006304 


.005077 


.004074 


5 


.008974 


.007234 


.005803 


.004636 


.003690 


6 


.008395 


.006715 


.005345 


.004235 


.003343 


7 


.007857 


.006237 


.004924 


.003870 


.003030 


8 


.007358 


.005795 


.004539 


.003538 


.002747 


9 


.006894 


.005387 


.004185 


.003236 


.002491 


40 


.006462 


.005009 


.003860 


.002960 


.002259 


1 


.006059 


.004660 


.003562 


.002708 


.002050 


2 


.005683 


.004336 


.003287 


.002478 


.001860 


3 


.005333 


.004036 


.003034 


.002268 


.001688 


4 


.005006 


.003958 


.002802 


.002077 


.001532 


5 


.004701 


.003500 


.002587 


.001902 


.001391 


6 


.004415 


.003260 


.002389 


.001742 


.001263 


7 


.004148 


.003045 


.002208 


.001595 


.001147 


8 


.003898 


.002831 


.002040 


.001461 


.001042 


9 


.003664 


.002638 


.001886 


.001339 


.0009459 


50 


.003444 


.002460 


.001743 


.001227 


.0008592 


1 


.003239 


.002294 


.001611 


.001124 


.0007805 


2 


.003046 


.002139 


.001490 


.001030 


.0007092 


3 


.002866 


.001995 


.001377 


.0009444 


.0006441 


4 


.002696 


.001861 


.001273 


.0008657 


.0005852 


5 


.002537 


.001736 


.001178 


.0007936 


.0005317 


6 


.002388 


.001620 


.001090 


.0007275 


.0004832 


7 


.002247 


.001512 


.001008 


.0006670 


.0004392 


8 


.002116 


'.001411 


.0009323 


.0006116 


.0003990 


9 


.001992 


.001317 


.0008625 


.0005608 


.0003626 


60 


.001876 


.001229 


.0007980 


0005142| 


.0003295 



CARLISLE EXPERIENCE. 



101 



TABLE NO. VII 
Carlisle Mortality Table 



X 

> 

CD 


2 

p 

< 
B' 
<n 


o 
& 

5' 


~ o 
o vj 


£1 CD 

ST. -i 
< O 

0<? i 


X 

> 
en 

CD 


p 
< 

B' 
en 


2 

p 

^ # 

B' 

en 


t- CD 

E-'P 
5f 2. 
o *< 

5' a 4 


El CD 

o «< 

HI <-i 
< O 

B$ 

en 7 




lx 


dx 


qx 


px 




lx 


dx 


qx 


px 





10000 


1539 


.15390 


.84610 


53 


4211 


68 


.01615 


.98385 


1 


8461 


682 


.08061 


.91939 


54 


4143 


70 


.01690 


.98310 


2 


7779 


505 


.06492 


.93508 


55 


4073 


73 


.01792 


.98208 


3 


7274 


276 


.03794 


.96206 


56 


4000 


76 


.01900 


.98100 


4 


6998 


201 


.02872 


.97128 


57 


3924 


82 


.02090 


.97910 


5 


6797 


121 


.01780 


.98220 


58 


3842 


93 


.02421 


.97579 


6 


6676 


82 


.01228 


.98772 


59 


3749 


106 


.02827 


.97173 


7 


6594 


58 


.00880 


.99120 


60 


3643 


122 


.03349 


.96651 


8 


6536 


43 


.00658 


.99342 


61 


3521 


126 


.03579 


.96421 


9 


6493 


33 


.00508 


.99492 


62 


3395 


127 


.03741 


.96259 


10 


6460 


29 


.00449 


.99551 


63 


3268 


125 


.03825 


.96175 


11 


6431 


31 


.00482 


.99518 


64 


3143 


125 


.03977 


.96023 


12 


6400 


32 


.00500 


.99500 


65 


3018 


124 


.04109 


.95891 


13 


6368 


33 


.00518 


.99482 


66 


2894 


123 


.04250 


.95749 


14 


6335 


35 


.00553 


.99447 


67 


2771 


123 


.04439 


.95561 


15 


6300 


39 


.00619 


.99381 


68 


2648 


123 


.04645 


.95355 


16 


6261 


42 


.00671 


.99329 


69 


2525 


124 


.04911 


.95089 


17 


6219 


43 


.00691 


.99309 


70 


2401 


124 


.05165 


.94835 


18 


6176 


43 


.00696 


.99304 


71 


2277 


134 


.05885 


.94115 


19 


6133 


43 


.00701 


.99299 


72 


2143 


146 


.06813 


.93187 


20 


6090 


43 


.00706 


.99294 


73 


1997 


156 


.07812 


.92188 


21 


6047 


42 


.00695 


.99305 


74 


1841 


166 


.09017 


.90983 


22 


6005 


42 


.00699 


.99301 


75 


1675 


160 


.09552 


.90448 


23 


5963 


42 


.00704 


.99296 


76 


1515 


156 


.10297 


.89703 


24 


5921 


42 


.00709 


.99291 


77 


1359 


146 


.10743 


.89257 


25 


5879 


43 


.00731 


.99269 


78 


1213 


132 


.10882 


.89118 


26 


5836 


43 


.00737 


.99263 


79 


1081 


128 


.11841 


.89159 


27 


5793 


45 


.00777 


.99223 


80 


953 


116 


.12172 


.87829 


28 


5748 


50 


.00870 


.99130 


81 


837 


112 


.13381 


.86619 


29 


5698 


56 


.00983 


.99017 


82 


725 


102 


.14069 


.85931 


30 


5642 


57 


.01010 


.98990 


83 


623 


94 


.15088 


.84912 


31 


5585 


57 


.01021 


.98979 


84 


529 


84 


.15879 


.84121 


32 


5528 


56 


.01013 


.98987 


85 


445 


78 


.17528 


.82472 


33 


5472 


55 


.01005 


.98995 


86 


367 


71 


.19346 


.80654 


34 


5417 


55 


.01015 


.98985 


87 


296 


64 


.21622 


.78378 


35 


5362 


55 


.01026 


.98974 


88 


232 


51 


.21983 


.78017 


36 


5307 


56 


.01055 


.98945 


89 


181 


39 


.21547 


.78453 


37 


5251 


57 


.01086 


.98914 


90 


142 


37 


.26056 


.73944 


38 


5194 


58 


.01117 


.98883 


91 


105 


30 


.28571 


.71429 


39 


5136 


61 


.01188 


.98812 


92 


75 


21" 


.28000 


.72000 


40 


5075 


66 


.01301 , 


.98699 


93 


54 


14 


.25926 


.74074 


41 


5009 


69 


.01378 


.98622 


94 


40 


10 


.25000 


.75000 


42 


4940 


71 


.01437 


.98563 


95 


30 


7 


.23333 


.76667 


43 


4869 


71 


.01458 


.98542 


96 


23 


5 


.21739 


.78261 


44 


4798 


71 


.01480 


.98520 


97 


18 


4 


.22222 


.77778 


45 


4727 


70 


.01481 


.98519 


98 


14 


3 


.21429 


.78571 


46 


4657 


69 


.01482 


.98518 


99 


11 


2 


.18182 


.81818 


47 


4588 


67 


.01460 


.98540 


100 


9 


2 


.22222 


.77778 


48 


4521 


63 


.01394 


.98606 


101 


7 


2 


.28571 


.71429 


49 


4458 


61 


.01368 


.98632 


102 


5 


2 


.40000 


.60000 


50 


4397 


59 


.01342 


.98658 


103 


3 


2 


.66667 


.33333 


51 


4338 


62 


.01429 


.98571 


104 


1 


1 


1.00000 


.00000 


52 


4276 


65 


.01520 


.98480 








i 





102 



FINANCE AND LIFE INSURANCE. 



TABLE NO. VIII. 
Northampton Mortality Table 





2 


Z 


cr k! 


s* 


X 


Z 


Z 


Sk; 


S^ 




o 


p 


E2 CD 


i— CD 




p 


p 


>— ■ a> 


►— CD 


X 


h 


& 






> 
era 




& 






> 


< 


**\ 


o<< 


o ^ 


CD 


< 


<<j 


o ^ 


o ^ 


CD 


B' 
era 


B' 

00 


& i-S 

^ o 
B' & 


2^ 
1-.. *-i 

< o 

p p 
era i 




B' 
era 


B' 
era 


3 cr 
era «f 


< O 

B'S 
era i 




lx 


dx 


qx 


px 




lx 


dx 


qx 


px 





11650 


3000 


.25751 


.74249 


49 


2936 


79 


.02691 


.97309 


1 


8650 


1367 


.15804 


.84196 


50 


2857 


81 


.02835 


.97165 


2 


7283 


502 


.06893 


.93107 


51 


2776 


82 


.02954 


.97046 


3 


6781 


335 


.04940 


.95060 


52 


2694 


82 


.03044 


.96956 


4 


6446 


197 


.03056 


.96944 


53 


2612 


82 


.03139 


.96861 


5 


6249 


184 


.02945 


.97055 


54 


2530 


82 


.03241 


.96759 


6 


6065 


140 


.02308 


.97692 


55 


2448 


82 


.03350 


.96650 


7 


5925 


110 


.01857 


.98143 


56 


2366 


82 


.03466 


.96534 


8 


5815 


80 


.01376 


.98624 


57 


2284 


82 


.03590 


.96410 


9 


5735 


60 


.01046 


.98954 


58 


2202 


82 


.03724 


.96276 


10 


5675 


52 


.00916 


.99084 


59 


2120 


82 


.03868 


.96132 


11 


5623 


50 


.00889 


.99111 


60 


2038 


82 


.04024 


.95976 


12 


5573 


50 


.00897 


.99103 


61 


1956 


82 


.04192 


.95808 


13 


5523 


50 


.00905 


.99095 


62 


1874 


81 


.04322 


.95678 


14 


5473 


50 


.00914 


.99086 


63 


1793 


81 


.04518 


.95482 


15 


5423 


50 


.00922 


.99078 


64 


1712 


80 


.04673 


.95327 


16 


5373 


53 


.00986 


.99014 


65 


1632 


80 


.04902 


.95098 


17 


5320 


58 


.01090 


.98910 


66 


1552 


80 


.05155 


94845 


18 


5262 


63 


.01197 


.98803 


67 


1472 


80 


.05435 


.94565 


19 


5199 


67 


.01289 


.98711 


68 


1392 


80 


.05747 


.94253 


20 


5132 


72 


.01403 


.98597 


69 


1312 


80 


.06098 


.93902 


21 


5060 


75 


.01482 


.98518 


70 


1232 


80 


.06493 


.93507 


22 


4985 


75 


.01505 


.98495 


71 


1152 


80 


.06944 


.93056 


23 


4910 


75 


.01528 


.98472 


72 


1072 


80 


:07463 


.92537 


24 


4835 


75 


.01551 


.98449 


73 


992 


80 


.08065 


.91935 


25 


4760 


75 


.01576 


.98424 


74 


912 


80 


.08772 


.91228 


26 


4685 


75 


.01601 


.98399 


75 


832 


80 


.09615 


.90385 


27 


4610 


75 


.01627 


.98373 


76 


752 


77 


.10239 


.89761 


28 


4535 


75 


.01654 


.98346 


77 


675 


73 


.10815 


.89185 


29 


4460 


75 


.01682 


.98318 


78 


602 


68 


.11296 


.88704 


30 


4385 


75 


.01710 


.98290 


79 


534 


65 


.12172 


.87828 


31 


4310 


75 


.01740 


.98260 


80 


469 


63 


.13433 


.86567 


32 


4235 


75 


.01771 


.98229 


81 


406 


60 


.14778 


.85222 


33 


4160 


75 


.01803 


.98197 


82 


346 


57 


.16474 


.83526 


34 


4085 


75 


.01836 


.98164 


83 


289 


55 


.19031 


.80969 


35 


4010 


75 


.01870 


.98130 


84 


234 


48 


.20513 


.79487 


36 


3935 


75 


.01906 


.98094 


85 


186 


41 


.22043 


.77957 


37 


3860 


75 


.01943 


.98057 


86 


145 


34 


.23448 


.76552 


38 


3785 


75 


.01981 


.98019 


87 


111 


28 


.25225 


.74775 


39 


3710 


75 


.02022 


.97978 


88 


83 


21 


.25301 


.74699 


40 


3635 


76 


.02091 


.97909 


89 


62 


16 


.25807 


.74193 


41 


3559 


77 


.02164 


.97836 


90 


46 


12 


.26087 


.73913 


42 


3482 


78 


.02240 


.97750 


91 


34 


10 


.29412 


.70588 


43 


3404 


78 


.02291 


.97709 


92 


24 


8 


.33333 


.66667 


44 


3326 


78 


.02345 


.97655 


93 


16 


7 


.43750 


.56250 


45 


3248 


78 


.02402 


.97598 


94 


9 


5 


.55556 


.44444 


46 


3170 


78 


.02461 


.97539 


95 


4 


3 


.75000 


.25000 


47 


3092 


78 


.02523 


.97477 


96 


1 


1 


1.00000 


[ .00000 


48 


3014 


78 


.02588 


.97412 










1 



ACTUARIES OK COMBINED EXPERIENCE. 



103 



TABLE NO. IX. 
Actuaries Mortality Table 



> 

era 

CD 


2 
p 

B' 
era 


3 

o 


E" CD 

*< o 


p— CD 

PS P 
o << 

< o 
0q 7 


X 

> 
era 

CD 


•z 

p 
< 
era 


3 

p 

5' 
era 


h- CD 

o << 

^ o 

a' CJ* 
era ? 


2**1 

!"2 CD 
PS & 

< o 
era i 




lx 


dx 


qx 


px 




lx 


dx 


qx 


px 


10 


100000 


676 


.00676 | 


.99324 


55 


63469 


1375 


| .02166 


.97834 


11 


99324 


674 


.00679 | 


.99321 


56 | 


62094 | 


1436 


| .02313 


.97687 


12 


98650 


672 


.00681 | 


.99319 


57 | 


60658 | 


1497 


| .02468 


.97532 


13 


97978 


671 


.00685 | 


.99315 


58 | 


59161 | 


1561 


| .02639 


.97361 


14 


97307 | 


671 


.00690 | 


.99310 


59 


57600 | 


1627 


| .02825 


.97175 


15 


96636 | 


671 


.00694 1 


.99306 


60 | 


55973 | 


1698 


| .03034 


.96966 


16 


95965 | 


672 


.00700 | 


.99300 


61 


54275 | 


1770 


| .03261 


.96739 


17 


95293 


673 


.00706 


.99294 


62 


52505 


1844 


| .03512 


.96488 


18 


94620 | 


675 


.00713 | 


.99287 


63 


50661 


1917 


| .03784 


.96216 


19 


93945 


677 


.00721 | 


.99279 


64 


48744 | 


1990 


| .04083 


.95917 


20 


93268 


680 


.00729 


.99271 


65 


46754 j 


2061 


| .04408 


.95592 


21 


92588 


683 


.00738 


.99262 


66 


44693 


2128 


| .04761 


.95239 


22 


91905 


685 


.00746 


.99254 


67 


42565 


2191 


| .05147 


.94853 


23 


91219 


690 


.00756 


.99244 


68 


40374 


2246 


| .05563 


.94437 


24 


90529 


694 


.00767 


.99233 


69 


3S128 


2291 


| .06009 


.93991 


25 


89835 


698 


.00777 


.99223 


70 


35837 


2327 


| .06493 


.93507 


26 


89137 


703 


.00789 


.99211 


71 


33510 


2357 


| .07016 


.92984 


27 


88434 


708 


.00801 


.99199 


72 


31159 


2362 


| .07580 


.92420 


28 


87726 


714 


.00814 


.99186 


73 


28797 


2358 


.08188 


.91812 


29 


87012 


720 


.00828 


.99172 


74 


26439 


2339 


| .08847 


.91153 


30 


86292 


727 


.00842 


.99158 


i 75 


24100 


2303 


| .09556 


.90444 


31 


85565 


734 


.00858 


.99142 


| 76 


21797 


2249 


| .10318 


.89682 


32 


84831 


742 


.00875 


.99125 


77 


19548 


2179 


| .11147 


.88853 


33 


84089 


750 


.00892 


.99108 


78 


17369 


2092 


| .12044 


.87956 


34 


83339 


758 


.00910 


.99090 


79 


15277 


1987 


| .13006 


.86994 


35 


82581 


767 


.00929 


.99071 


80 


13290 


1866 


| .14041 


.85957 


36 


81814 


776 


.00948 


.99052 


81 


11424 


1730 


| .15144 


.84856 


37 


81038 


785 


.00969 


.99031 


82 


9694 


1582 


| .16319 


.83681 


38 


80253 


795 


.00991 


.99009 


83 


8112 


1427 


| .17591 


.82409 


39 


79458 


805 


.01013 


.98987 


84 


6685 


1268 


1 .18968 


.81032 


40 


78653 


815 


.01036 


.98964 


85 


5417 


1111 


| .20510 


.79490 


41 


77838 


826 


.01061 


.98939 


86 


4306 


958 


1 .22248 


.77752 


42 


77012 


839 


.01089 


.98911 


87 


3348 


811 


1 .24223 


.75777 


43 


76173 


857 


.01125 


.98875 


88 


2537 


673 


| .26527 


.73473 


44 


75316 


881 


.01170 


.98830 


89 


1864 


545 


| .29238 


.70762 


45 


74435 


909 


.01221 


.98779 


90 


1319 


427 


| .32373 


.67627 


46 


73526 


944 


.01284 


.98716 


91 


892 


322 


| .36099 


.63901 


67 


72582 


981 


.01352 


.98648 


92 


570 


231 


| .40526 


.59474 


48 


71601 


1021 


.01426 


.98574 


93 


339 


155 


| .45723 


.54277 


49 


70580 


1063 


.01506 


1 .98494 


94 


184 


95 


| .51630 


.48370 


50 


69517 


1108 


.01594 


.98406 


95 


89 


52 


| .58427 


.41573 


51 


68409 


1156 


.01690 


.98310 


96 


37 


24 


| .64865 


.35135 


52 


67253 


1207 


.01795 


.98205 


97 


13 


9 


| .69231 


.30769 


53 


66046 


1261 


.01909 


.98091 


98 


4 


3 


.75000 


.25000 


54 


64785 


1316 


.02031 


.97969 


99 


1 


1 


11.00000 


.00000 



104 



FINANCE AND LIFE INSURANCE. 



TABLE NO. IX. 



Actuaries Mortality Table — Makehamized 



> 

era 

CD 


2 

p 

< 
B 

era 


2 

o 

B' 
era 


t-. CD 

sg 

o ^ 

*< O 

era 7 


o *< 

m. >-i 

< O 

a * 
crq 7 


X 

> 

QTQ 

CD 


25 

p 

B' 
era 


3 

p 

p- 

B' 


E2 CD 

1-15 *a 

< o 

P Q3 

crq 7 


t- CD 

*< o 
crq 7 




lx 


dx 


qx 


px 




lx 


dx 


qx 


px 


10 


100030 


691 


.99310 


.00690 


57 


60504 


1487 


.97543 


.02457 


11 


99339 


688 


.99307 


.00693 


58 


59017 


1549 


.97375 


.02625 


12 


98651 


687 


.99304 


.00696 


59 


57468 


1614 


.97191 


.02809 


13 


97964 


684 


.99301 


.00699 


60 


55854 


1682 


.96990 


.03010 


14 


97280 


683 


.99298 


.00702 


61 


54172 


1750 


| .96770 


.03230 


15 


96597 


681 


.99295 


.00705 


62 


52422 


1818 


.96530 


.03470 


16 


95916 


680 


.99291 


.00709 


63 


50604 


1889 


.96263 


.03832 


17 


95236 


679 


.99287 


.00712 


64 


48715 


1957 


.95982 


.04018 


18 


94557 


679 


.99282 


.00718 


65 


46758 


2024 


.95670 


.04230 


19 


93878 


679 


.99277 


.00723 


66 


44734 


2090 


.95330 


.04670 


20 


93199 


678 


.99272 


.00728 


67 


42644 


2150 


.94958 


.05042 


21 


92521 


680 


.99266 


.00734 


68 


40494 


2206 


.94553 


.05447 


22 


91841 


681 


.99259 


.00741 


69 


38288 


2254 j .94111 


.05889 


23 


91161 


682 


.99252 


.00748 


70 


36034 


2296 


| .93630 


.06370 


24 


90479 


684 


.99244 


.00756 


71 


33738 


2326 


| .93106 


.06894 


25 


89795 


686 


.99236 


.00768 


72 


31412 


2344 


| .92536 


.07464 


26 


89109 


690 


.99226 


.00774 


73 


29068 


2351 


| .91915 


.08085 


27 


88419 


693 


.99216 


.00784 


74 


26717 


2340 


| .91240 


.08760 


28 


87726 


698 


.99205 


.00795 


75 


24377 


2314 


| .90507 


.09493 


29 


87028 


703 


.99192 


.00808 


76 


22063 


2270 


| .89712 


.10288 


30 


86325 


709 


.99179 


.00821 


77 


19792 


2207 


I .88848 


.11152 


31 


85616 


716 


.99164 


.00836 


78 


17596 


2126 


| .87913 


.12087 


32 


84900 


724 


.99148 


.00852 


79 


15460 


2025 


| .86900 


.12100 


33 


84176 


732 


.99130 


.00870 


80 


13425 


1907 


| .85804 


.14196 


34 


83444 


743 


.99110 


.00890 


81 


11528 


1773 


| .84620 


.15380 


35 


82701 


753 


.99089 


.00911 


82 


9755 


1625 


.83342 


.16658 


36 


81948 


765 


.99066 


.00934 


83 


8130 


1464 


| .81965 


.18035 


37 


81183 


780 


.99040 


.00960 


84 


6666 


1303 


| .80484 


.19516 


38 


80403 


794 


.99012 


.00988 


85 


5363 


1132 


| .78893 


.21107 


39 


79609 


810 


.98982 


.01018 


86 


4231 


965 


I .77187 


.22813 


40 


78799 


829 


.98948 


.01052 


87 


3266 


805 


| .75361 


.24639 


41 


77970 


848 


.98912 


.01088 


88 


2461 


654 


| .73411 


.26589 


42 


77122 


870 


.98872 


.01128 


89 


1807 


518 


| .71333 


.28667 


43 


76252 


894 


.98828 


.01172 


90 


1289 


398 


1 .69128 


.30872 


44 


75358 


919 


.98780 


.01270 


91 


891 


296 


1 .66790 


.33210 


45 


74439 


948 


.98727 


.01273 


92 


595 


212 


| .64321 


.35679 


46 


73491 


977 


.98670 


.01330 


93 


383 


147 


.61723 


.38277 


47 


72514 


1010 


.98607 


.01393 


94 


236 


97 


.58998 


.41001 


48 


71504 


1045 


.98538 


.01462 


95 


139 


61 


.56152 


.43848 


49 


70459 


1083 


.98463 


.01537 


96 


78 


36 


I .53193 


.46807 


50 


69376 


1123 


.98381 


.01618 


97 


42 


21 


| .50132 


.49868 


51 


68253 


1167 


.98291 


.01709 


98 


21 


11 


| .46983 


.53017 


52 


67086 


1212 


.98192 


.01808 


99 


10 


6 


.43760 


.56240 


53 


65874 


1262 


.98085 


.01915 


100 


4 


2 


1 .37180 


.62820 


54 


64612 


1314 


.97967 


.02023 


101 


2 


1 


1 .33869 


.66131 


55 


63298 


1386 


.97838 


.02162 


102 


1 


1 


| .00000 


11.00000 


56 


61930 


1426 


.97697 


.02303 








1 


1 



AMERICAN EXPERIENCE. 



105 



TABLE NO. X. 



American Experience Mortality Table. 



X 

> 

0> 


p 


o 
5' 

0q 


2 ^ 

& CO 

<< a 

o «< 

<< o 
5' cr 


& o 

£a 

<* • 


> 
org 

CO 


2 
p 

B' 


2 

p 


E-- CD 

S-'P 

«j a 

o ^ 

^ hrt 

«< O 

5" cr 


~ CO 

S P 

<£a 
o *< 

< o 
or? i 




lx 


dx 


qx 


px 




lx 


dx 


qx 


px 


10 


100000 


749 


.00749 


.99251 


53 


66797 


1091 


.01633 


.98367 


11 


99251 


746 


.00752 


.99248 


54 


65706 


1143 


.01740 


.98260 


12 


98505 


743 


.00754 


.99246 


55 


64563 


1199 


.01857 


.98143 


13 


97762 


740 


.00757 


.99343 


56 


63364 


1260 


.01988 


.98012 


14 


97022 


737 


.00760 


.99240 


57 


62104 


1325 


.02134 


.97866 


15 


96285 


735 


.00763 


.99237 


58 


60779 


1394 


.02294 


.97706 


16 


95550 


732 


.00766 


.99234 


59 


59385 


1468 


I .02472 


.97528 


17 


94818 


729 


.00769 


.99231 


60 


57917 


1546 


.02669 


.97331 


18 


94089 


727 


.00773 


.99227 


61 


56371 


1628 


.02888 


.97112 


19 


93362 


725 


.00777 


.99223 


62 


54743 


1713 


.03129 


.96871 


20 


92637 


723 


.00781 


.99219 


63 


53030 


1800 


.03394 


.96606 


21 


91914 


722 


.00786 


.99214 


64 


51230 


1889 


.03687 


.96313 


22 


91192 


721 


.00791 


.99209 


65 


49341 


1980 


.04013 


.95987 


23 


90471 


720 


.00796 


.99204 


66 


47361 


2070 


.04371 


.95629 


24 


89751 


719 


.00801 


.99199 


67 


45291 


2158 


.04765 


.95235 


25 


89032 


718 


.00807 


.99193 


68 


43133 


2243 


.05200 


.94800 


26 


88314 


718 


.00813 


.99187 


69 


40890 


2321 


.05676 


.94324 


27 


87596 


718 


.00820 


.99180 


70 


38569 


2391 


.06199 


.93801 


28 


86878 


718 


.00826 


.99174 


71 


36178 


2448 


.06767 


.93233 


29 


86160 


719 


.00835 


.99165 


72 


33730 


2487 


.07373 


.92627 


30 


85441 


720 


.00843 


.99157 


73 


31243 


2505 


.08018 


.91982 


31 


84721 


721 


.00851 


.99149 


74 


28738 


2501 


.08703 


.91297 


32 


84000 


723 


.00861 


.99139 


75 


26237 


2476 


.09437 


' .90563 


33 


83277 


726 


.00872 


.99128 


76 


23761 


2431 


.10231 


.89769 


34 


82551 


729 


.00883 


.99117 


77 


21330 


2369 


.11106 


.88894 


35 


81822 


732 


.00895 


.99105 


78 


18961 


2291 


.12083 


.87917 


36 


81090 


737 


.00909 


.99091 


79 


16670 


2196 


.13173 


.86827 


37 


80353 


742 


.00923 


.99077 


80 


14474 


2091 


.14447 


.85553 


38 


79611 


749 


.00941 


.99059 


81 


12383 


1964 


.15861 


.84139 


39 


78862 


756 


.00959 


.99041 


82 


10419 


1816 


.17430 


.82570 


40 


78106 


765 


.00979 


.99021 


83 


8603 


1648 


.19156 


.80844 


41 


77341 


774 


.01001 


.98999 


84 


6955 


1470 


.21136 


.78864 


42 


76567 


785 


.01025 


.98975 


85 


5485 


1292 


.23555 


.76445 


43 


75782 


797 


.01052 


.98948 


86 


4193 


1114 


.26568 


.73432 


44 


77985 


812 


.01083 


.98917 


87 


3079 


933 


.30302 


.69698 


45 


74173 


828 


.01116 


.98884 


88 


2146 


744 


.34669 


.65331 


46 


73345 


848 


.01158 


.98844 


89 


1402 


555 


.39586 


.60414 


47 


72497 


870 


.01200 


.98800 


90 


847 


385 


.45455 


.54545 


48 


71627 


896 


.01251 


.98749 


91 


462 


246 


.53247 


.46753 


49 


70731 


927 


.01311 


.98689 


92 


216 


137 


.63426 


.36574 


50 


69804 


762 


.01378 


.98622 


93 


79 


58 


.73418 


.26582 


51 


68842 


1001 


.01454 


| .98546 


94 


21 


18 


.85714 


.14286 


52 


67842 


1041 


.01539 


.98461 


95 


3 


3 


1.00000 


.00000 



106 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XI. 

National Fraternal Congress Table 
of Mortality. 



X 

> 


OKI g 

o 4 

1 


v- x 
era g 

B 

a 1 

CD 


qx Yearly 
Probability 
of Dying 


1000 qx 
Yearly Death 
Rate Per 1000 


1000 qxv Dis- 
counted yearly 
Rate Per 1000 


O X 

Ms <-T 
1 


O <! 

O QJ 

S 

P3 




20 


100000 


500 


.0050000 


5.000 


4.808 


45.6 


49.7 


20.4916 


21 


99500 


501 


.0050352 


5.035 


4.841 


44.9 


48.8 


20.3731 


22 


98999 


502 


.0050708 


5.071 


4.876 


44.1 


47.9 


20.2500 


23 


98497 


503 


.0051068 


5.107 


4.911 


43.3 


47.0 


20.1220 


24 


97994 


505 


.0051535 


5.154 


4.956 


42.5 


46.1 


19.9890 


25 


97489 


507 


.0052006 


5.201 


5.001 


41.8 


45.2 


19.8508 


26 


96982 


510 


.0052587 


5.259 


5.057 


41.0 


44.3 


19.7074 


27 


96472 


513 


.0053176 


5.318 


5.113 


40.2 


43.4 


19.5585 


28 


95959 


517 


.0053877 


5.388 


5.181 


39.4 


42.5 


19.4040 


29 


95442 


522 


.0054693 


5.469 


5.259 


38.6 


41.6 


19.2439 


30 


94920 


527 


.0055520 


5.552 


5.338 


37.8 


40.7 


19.0780 


31 


94393 


533 


.0056466 


5.647 


5.430 


37.0 


39.8 


18.9060 


32 


93860 


540 


.0057532 


5.753 


5.532 


36.2 


38.9 


18.7280 


33 


93320 


548 


.0058723 


5.872 


5.646 


35.4 


38.0 


18.5439 


34 


92772 


557 


.0060040 


6.004 


5.773 


34.6 


37.1 


18.3534 


35 


92215 


567 


.0061487 


6.149 


5.912 


33.9 


36.2 


18.1565 


36 


91648 


578 


.0063067 


6.307 


6.064 


33.1 


35.3 


17.9532 


37 


91070 


591 


.0064895 


6.490 


6.240 


32.3 


34.4 


17.7432 


38 


90479 


606 


.0066977 


6.698 


6.440 


31.5 


33.5 


17.5267 


39 


89873 


622 


.0069209 


6.921 


6.655 


30.7 


32.6 


17.3036 


40 


89251 


640 


.0071708 


7.171 


6.895 


29.9 


31.7 


17.0740 


41 


88611 


660 


.0074483 


7.448 


7.162 


29.1 


30.9 


16.8376 


42 


87951 


683 


.0077657 


7.766 


7.467 


28.3 


30.0 


16.5948 


43 


87268 


708 


.0081129 


8.113 


7.801 


27.5 


29.1 


16.3455 


44 


86560 


734 


.0084797 


8.480 


8.154 


26.8 


28.2 


16.0898 


45 


85826 


761 


.0088668 


8.867 


8.526 


26.0 


27.4 


15.8276 


46 


85065 


790 


.0092870 


9.287 


8.930 


25.2 


26.5 


15.5587 


47 


84275 


822 


.0097538 


9.754 


9.379 


24.4 


25.6 


15.2830 


48 


83453 


857 


.0102693 


10.269 


9.874 


23.7 


24.8 


15.0006 


49 


82596 


894 


.0108238 


10.824 


10.408 


22.9 


23.9 


14.7117 


50 


81702 


935 


.0114440 


11.444 


11.004 


22.2 


23.1 


14.4162 


51 


80767 


981 


.0121460 


12.146 


11.679 


21.4 


22.2 


14.1144 


52 


79786 


1029 


.0128970 


12.89T 


12.401 


20.7 


21.4 


13.8067 


53 


78757 


1083 


.0137512 


13.751 


13.222 


19.9 


20.6 


13.4930 


54 


77674 


1140 


.0146767 


14.677 


14.112 


19.2 


19.8 


13.1738 


55 


76534 


1202 


.0157054 


15.705 


15.101 


18.5 


19.0 


12.8494 


56 


75332 


1270 


.0168587 


16.859 


16.211 


17.8 


18.2 


12.5200 


57 


74062 


1342 


.0181200 


18.120 


17.423 


17.1 


17.4 


12.1862 


58 


72720 


1418 


.0194994 


19.499 


18.749 


16.4 


16.6 


11.8484 


59 


71302 


1501 


.0210513 


21.051 


20.241 


15.7 


15.8 


11.5067 



FRATERNAL CONGRESS. 



107 



TABLE NO. XL (Concluded) 

National Fraternal Congress Table 
of Mortality. 



•< 


P £ 


— Oi 


o ^j>a 


*? ^ - 1 


22 £ 


C <x> 


a > 


o> 


>< 




«< x 


■-* >s . X 


03 CD 2 


p° 2 


o x 


2 < 


C 3 


crq • 

CD 


2. Z 

CD 
*1 


<*» £ 

3 

c 

CD 


Yearly 

obabiity 

Dying 


)0 qx 

arly Death 

te Per 1000 


)0 qx v Dis- 
inted Yearly 
te Per 1000 


CD p 


3 CD 
°* 

-i 

P 


CD C3 
rt- 

(l+O 


60 


69801 


1588 


.0227504 


22.750 


21.875 


15.0 


15.1 


11.1619 


61 


68213 


1681 


.0246434 


24.643 


23.695 


14.4 


14.4 


10.8144 


62 


66532 


1778 


.0267240 


26.724 


25.696 


13.7 


13.6 


10.4649 


63 


64754 


1880 


.0290330 


29.033 


27.916 


13.1 


12.9 


10.1138 


64 


62S74 


1985 


.0315711 


31.571 


30.357 


12.4 


12.2 


9.7617 


65 


60889 


2094 


.0343904 


34.390 


33.067 


11.8 


11.6 


9.4092 


66 


58795 


2206 


.0375202 


37.520 


36.077 


11.2 


10.9 


9.0571 


67 


56589 


2318 


.0409620 


40.962 


39.387 


10.7 


10.3 


8.7060 


68 


54271 


2430 


.0447753 


44.775 


43.053 


10.1 


9.7 


8.3566 


69 


51841 


2539 


.0489767 


48.977 


47.093 


9.5 


9.1 


8.0095 


70 


49302 


2645 


.0536489 


53.649 


51.586 


9.0 


8.5 


7.6653 


71 


46657 


2744 


.0588122 


58.812 


56.550 


8.5 


7.9 


7.3248 


72 


43913 


2832 


.0644912 


64.491 


62.011 


8.0 


7.4 


6.9889 


73 


41081 


2909 


.0708113 


70.811 


68.087 


7.5 


6.9 


6.6578 


74 


38172 


2969 


.0777795 


77.780 


74.788 


7.0 


6.4 


6.3325 


75 


45203 


3009 


.0854757 


85.476 


82.188 


6.6 


6.0 


6.0135 


76 


32194 


3026 


.0939927 


93.993 


90.377 


6.2 


5.5 


5.7014 


77 


29168 


3016 


.1034010 


103.401 


99.424 


5.7 


5.1 


5.3967 


78 


26152 


2977 


.1138345 


113.835 


109.457 


5.3 


4.7 


5.0999 


79 


23175 
20270 


2905 


.1253506 


125.351 


120.530 


5.0 


4.3 


4.8117 


80 


2799 


.1380858 


138.086 


132.775 


4.6 


4.0 


4.5323 


81 


17471 


2659 


.1521951 


152.195 


146.341 


4.3 


3.6 


4.2621 


82 


14812 


2485 


.1677694 


167.769 


161.316 


3.9 


3.3 


4.0016 


83 


12327 


2280 


.1849599 


184.960 


177.846 


3.6 


3.0 


3.7510 


84 


10047 


2050 


.2040410 


204.041 


196.193 


3.3 


2.8 


3.5102 


85 


7997 


1800 


.2250844 


225.084 


216.427 


3.0 


2.5 


3.2799 


86 


6197 


1539 


.2483460 


248.346 


238.794 


2.8 


2.3 


3.0598 


87 


4658 


1277 


.2741520 


274.152 


263.608 


2.5 


2.0 


2.8500 


88 


3381 


1023 


.3025732 


302.573 


290.035 


2.3 


1.8 


2.6507 


89 


2358 


788 


.3341815 


334.182 


321.329 


2.1 


1.7 


2.4614 


90 


1570 


579 


.3687898 


368.790 


354.606 


1.9 


1.5 


2.2828 


91 


991 


404 


.4076690 


407.669 


391.989 


1.7 


1.4 


2.1135 


92 


587 


264 


.4497445 


449.745 


432.447 


1.5 


1.2 


1.9551 


93 


323 


161 


.4984520 


498.452 


479.281 


1.4 


1.0 


1.8051 


94 


162 


89 


.5493827 


549.383 


528.253 


1.2 


.9 


1.6695 


95 


73 


44 


.6027397 


602.740 


579.557 


1.1 


.8 


1.5452 


06 


29 


19 


.6551724 


655.172 


629.973 


1.0 


.8 


1.4272 


97 


10 


7 


.7000000 


700.000 


673.077 


.8 


.7 


1.2885 


98 


3 


3 


1.0000000 


1000.000 


961.538 


.5 


.5 


1.0000 



108 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XII. 

Commutation Columns for Annuities 
Actuariescor Combined Experience. 
One Life. 



X 


Q*| 


O^ 


am 


OCX! 




CD H' 


CD *-"' 


CD K* 


CD tt" 


> 


3 


* $ 


p X 


P X 


<3 


r* CD 


c+- © 


?+ T) 


* ^ 


a> 


^d 


hj 


CD 


CD 




CD 
"J 


CD 


>-t 


*-l 




Dx 


Kx 


Dx 


Nx 


10 


61391.32 


1016386.90 


55839.49 


801152.40 


11 


58072.68 


958314.22 


52322.65 


748829.75 


12 


54932.01 


903382.21 


49026.02 


699803.73 


13 


51959.83 


851422.38 


45935.92 


653867.81 


14 


49146.65 


802275.73 


43038.98 


610828.83 


15 


46483.57 


755792.16 


40322.83 


570506.00 


16 


43962.68 


711829.48 


37776.27 


532729.73 


17 


41576.03 


670253.45 


35388.43 


497341.30 


18 


39316.56 


630936.89 


33149.53 


464191.77 


19 


37177.23 


593759.66 


31050.04 


433141.73 


20 


35151.73 


558607.93 


29081.41 


404060.32 


21 


33233.76 


525374.17 


27235.26 


376825.06 


22 


31417.71 


493956.46 


25504.11 


351320.95 


23 


29698.29 


464258.17 


23880.83 


327440.12 


24 


28070.14 


436188.03 


22358.72 


305081.40 


25 


26528.52 


409659.51 


20931.43 


284149.97 


26 


25068.95 


384590.56 


19593.21 


264556.76 


27 


23686.90 


360963.66 


18338.38 


246218.38 


28 


22378.34 


338525.32 


17161.85 


229056.53 


29 


21139.25 


317386.07 


16058.65 


212997.88 


30 


19966.02 


297420.05 


15024.31 


197073.57 


31 


18855.06 


278564.99 


14054.47 


183919.10 


32 


17803.16 


260761.83 


13145.19 


170773.91 


33 


16807.08 


243954.75 


12292.65 


158481.26 


34 


15863.97 


228090.78 


11493.40 


146987.86 


35 


14971.13 


213119.65 


10744.22 


136243.64 


36 


14125.79 


198993.86 


10041.91 


126201.733 


37 


13325.53 


185668.33 


9383.648 


116818.085 


38 


12568.05 


173100.28 


8766.746 


108051.339 


39 


11851.00 


161249.28 


8188.585 


99862.754 


40 


11172.32 


150076.96 


7646.818 


92215.936 


41 


10530.05 


139546.906 


7139.228 


85076.708 


42 


9922.196 


129624.710 


6663.618 


78413.060 


43 


9346.763 


120277.947 


6217.974 


72195.086 


44 


8801.527 


111476.420 


5800.015 


66395.071 


45 


5284.357 


103192.063 


5407.775 


60987.296 


46 


7793.512 


95398.551 


5033.512 


55953.784 


47 


7327.695 


88071.456 


4693.028 


51260.756 


48 


6883.872 


81187.584 


4367.546 


46893.210 


49 


6462.582 


74725.002 


4061.573 


42831.637 


50 


6062.142 


68662.860 


3773.963 


39057.674 


51 


5681.468 


62981.392 


3503.597 


35554.071 


52 


5319.447 


57661.945 


3249.426 


32304.651 


53 


4975.236 


52686.709 


3010.479 


29294.172 


54 


4647.552 


48038.857 


2785.851 


26508.321 



SINGLE LIFE COLUMNS, 



109 



TABLE NO. XII. (Concluded) 

Commutation Columns for Annuities. 
Actuaries or Combined Experience. 
One Life. 



> 

OTQ 


Five Per 
Cent 


C-3 


aw 
g x 


a rr 




Dx 


Xx 


Dx 


Nx 


55 


4336.607 


43702.248 


2574.774 


23933.547 


56 


4040.625 


39661.623 


2376.409 


21557.138 


57 


3759.223 


35902.400 


2190.048 


19367.090 


58 


3491.854 


32410.546 


2015.094 


17351.996 


59 


3237.828 


29172,718 


1850.872 


15501.124 


60 


2996.544 


26176.174 


1696.784 


13804.340 


61 


2767.277 


23408.897 


1552.180 


12252.160 


62 


2549.554 


20859.343 


1416.567 


10835.593 


63 


2342.869 


18516.474 


1289.449 


9546.144 


64 


2146.871 


16369.603 


1170.431 


8375.713 


65 


1961.166 


14408.437 


1059.101 


7316.612 


66 


1785.442 


12622.995 


955.1076 


6361.504 


67 


1619.458 


11003.537 


858.1431 


5503.361 


68 


1462.950 


9540.587 


767.8968 


4735.464 


69 


1315.777 


8224.810 


684.1311 


4051.333 


70 


1177.825 


7046.985 


606.6261 


3444.707 


71 


1048.900 


5998.085 


535.1283 


2909.579 


72 


928.8678 


5069.2172 


469.4197 


2440.159 


73 


817.5765 


4251.6407 


409.2785 


2030.880 


74 


714.8862 


3536.7545 


354.4957 


1676.384 


75 


620.6111 


2916.1434 


304.8436 


1371.540 


76 


534.5765 


2381.5669 


260.1064 


1111.434 


77 


456.5900 


1924.9769 


220.0649 


891.369 


78 


386.3754 


1538.6015 


184.4664 


706.903 


79 


323.6558 


1214.9457 


153.0645 


553.838 


80 


268.1521 


946.7936 


125.6191 


428.219 


81 


219.5255 


727.2681 


101.8691 


326.3498 


82 


177.4109 


549.8572 


81.5496 


244.8002 


83 


141.3891 


408.4681 


64.3785 


180.4217 


84 


110.9686 


297.4995 


50.0505 


130.3712 


85 


85.6384 


211.8611 


38.2614 


92.1098 


86 


64.8327 


147.0284 


28.6926 


63.4172 


87 


48.0083 


99.0201 


21.0463 


42.3709 


88 


34.6467 


64.3733 


15.0454 


27.3255 


89 


24.2437 


40.1297 


10.4286 


16.8970 


90 


16.3383 


23.7913 


6.9617 


9.9352 


91 


10.5230 


13.2684 


4.4415 


5.4937 


92 


6.4041 


6.8642 


2.6775 


2.8162 


93 


3.6273 


3.2369 


1.5023 


1.3139 


94 


1.8751 


1.3618 


.7692 


.5446 


95 


.8638 


.4980 


.3510 


.1936 


96 


.3420 


.1560 


.1377 


.0560 


97 


.1144 


.0415 


.0456 


.0103 


98 


.0335 


.0080 


.0103 


.0000 


99 


1 .0080 


| .0000 


1 





110 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XIII. 

Carlisle Table. 
Annuity Commutation Columns. One Life. 



* 

> 

ere 

CD 


Q *3 

CD 


Q *1 

CD ►* 
CD 


Q02 

§ R" 

CD 


g.R 

CD 


> 
ere 

CD 




Dx 


Nx 


Dx 


Nx 




1 


8058.094 


112772.078 


7982.076 


96415.031 


1 


2 


7055.767 


105716.311 


6923.282 


89491.749 


2 


3 


6283.555 


99432.756 . 


6107.390 


83384.359 


3 


4 


5757.271 


93675.485 


5543.070 


77841.289 


4 


5 


5325.628 


88349.857 


5079.112 


72762.177 


5 


6 


4981.733 


83368.124 


4706.316 


68055.861 


6 


7 


4686.232 


78681.892 


4385.385 


63670.476 


7 


8 


4423.821 


74258.071 


4100.767 


59569.709 . 


8 


9 


4185.445 


70072.626 


3843.196 


55726.513 


9 


10 


3965.879 


66106.747 


3607.229 


52119.284 


10 


11 


3760.078 


62346.669 


3387.770 


48731.514 


11 


12 


3563.760 


58782.909 . 


3180.604 


45550.910 


12 


13 


3377.086 


55405.823 


2985.567 


42565.343 


13 


14 


3199.605 


52206.218 


2801.976 


39763.367 


14 


15 


3030.407 


49175.811 


2628.770 


37134.597 


15 


16 


2868.236 


46307.575 


2464.619 


34669.978 


16 


17 


2713.330 


43594.245 


2309.515 


32360.463 


17 


18 


2566.235 


41028.010 


2163.723 


30196.740 


18 


19 


2427.036 


38600.974 


2027.036 


28169.704 


19 


20 


2295.256 


36305.718 


1898.890 


26270.814 


20 


21 


2170.525 


34135.193 


1778.757 


24492.057 


21 


22 


2052.804 


32082.389 


1666.418 


22825.639 


22 


23 


1941.381 


30141.008 


1561.097 . 


21264.542 


23 


24 


1835.912 


28305.096 


1462.360 


19802.182 


24 


25 


1736.085 


26569.011 


1369.799 


18432.383 


25 


26 


1641.320 


24927.691 


| 1282.810 


17149.573 


26 


27 


1551.645 


23376.046 


1 1201.282 


15948.291. 


27 


28 


1466.278 


21909.768 


1 1124.482 


14823.809 


28 


29 


1384.308 


20525.460 


1 1051.604 


13772.2048 


29 


30 


1305.431 


19220.029 


982.3293 


12789.8755 


30 


31 


1230.707 


17989.322 


917.3634 


11872.5121 


31 


32 


1160.140 


16829.182 


856.6044 


11015.9077 


32 


33 


1093.702 


15735.480 


799.9311 


10215.9766 


33 


34 


1031.151 


14704.3291 


747.0667 


9468.9099 


34 


35 


972.0784 


13732.2507 


| 697.6242 


8771.2857 


35 


36 


916.2929 


12815.9578 


651.3853 


8119.9004 


36 


37 


863.4514 


11952.5064 


608.0300 


7511.8704 


37 


38 


813.4081 


11139.0983 


567.3864 


6944.4840 


38 


39 


766.0238 


10373.0745 \ 


529.2931 


6415.1909 


39 


40 


720.8816 


9652.1929 | 


493.4026 


5921.7883 


40 


41 


677.6254 


8974.5675 | 


459.4206 


5462.3677 


41 


42 


636.4676 


8338.0999 | 


427.4452 


5034.9225 


42 


43 


597.4476 


7740.6523 | 


397.4546 


4637.4679 


43 


44 


560.7005 


7179.9518 | 


369.4895 


4267.9784 


44 


45 


526.0986 


6653.8532 1 


343.4168 


3924.5616 


45 


46 


493.6265 


6160.2267 | 


319.1805 


3605.3811 


46 


47 


463.1548 


5697.0719 


296.6522 


3308.7289 


47 


48 


434.6585 


5262.4134 1 


275.7737 


3032.9552 


48 


49 


408.1919 


4854.2215 | 


256.5385 


2776.4167 


49 


50 


383.4349 


4470.7866 


238.7059 


2537.7108 


50 


51 


360.2757 


4110.5109 


222.1725 


2315.5383 


51 


52 


338.2159 


3772.2950 


206.6011 


2108.9372 


52 



SINGLE LIFE COLUMNS. 



Ill 



TABLE NO. XIII. (Concuded) 

Carlisle Table. 

Annuity Commutation Columns. One Life. 



> 

or? 

CD 


CD H* 

CD 

>-i 


CD 5" 

CD 


§ R 

CD 


~-d 

CD 
-i 


X 

> 

CD 




Dx 


Nx 


Dx 


Nx 




53 


317.2140 


3455.0810 


191.9439 


1916.9933 


53 


54 


297.2300 


3157.8510 


178.1550 


1738.8383 


54 


55 


278.2934 


2879.5576 


165.2311 


1573.6072 


55 


56 


260.2910 


2619.2666 


153.0845 


1420.5227 


56 


57 


243.1861 


2376.0805 


141.6754 


1278.8473 


57 


58 


226.7660 


2149.3145 


130.8631 


1147.9842 


58 


59 


210.7398 


1938.5747 


120.4674 


1027.5168 


59 


60 


195.0299 


1743.5448 


110.4351 


917.0817 


60 


61 


179.5224 


1564.0224 


100.6951 


816.38658 


61 


62 


164.8554 


1399.1670 


91.59592 


724.79066 


62 


63 


151.1319 


1248.0351 


83.17877 


641.61189 


63 


64 


138.4293 


1109.6058 


75.46905 


566.14284* 


64 


65 


126.5944 


983.0114 


68.36566 


497.77718 


65 


66 


115.6124 


867.3990 


61.84596 


435.93122 


66 


67 


105.4297 


761.96934 


55.86549 


380.06573 


67 


68 


95.95015 


666.01919 


50.36388 


329.70185 


68 


69 


87.13644 


578.88275 


45.30610 


284.39575 


69 


70 


78.91167 


499.97108 


40.64261 


243.75314 


70 


71 


71.27262 


428.69846 


36.36190 


207.39124 


71 


72 


63.88407 


364.81439 


32.28496 


175.10628 


72 


73 


56.69689 


308.11750 


28.38244 


146.72384 


73 


74 


49.77894 


258.33856 


24.68424 


122.03960 


74 


75 


43.13377 


215.20479 


21.18726 


100.85234 


75 


76 


37.15572 


178.04907 


18.07870 


82.77364 


76 


77 


31.74267 


146.30640 


15.29918 


67.47446 


77 


78 


26.98333 


119.32307 


12.88260 


54.59186 


78 


79 


22.90187 


96.42120 


10.83084 


43.76102 


79 


80 


19.22866 


77.19254 


9.00790 


34.75312 


80 


81 


16.08393 


61.10861 


7.46364 


27.28948 


81 


82 


13.26830 


47.84031 


6.09898 


21.19050 


82 


83 


10.85866 


36.98165 


4.94426 


16.24623 


83 


84 


8.78121 


28.20044 


3.96062 


12.28561 


84 


85 


7.03509 


21.16535 | 


3.14312 


9.14249 


85 


86 


5.52569 


15.63966 


2.44547 


6.69702 


86 


87 


4.24446 


11.39520 


1.86072 


4.83630 


87 


88 


3.16832 


8.22688 


1.37585 


3.46045 


88 


89 


2.35413 


5.87275 


1.01264 


2.44781 


89 


90 


1.75894 


4.11381 


.74948 


1.69833 


90 


91 


1.23869 


2.87512 


.52282 


1.17551 


91 


92 


.84265 


2.03247 


.35231 


.82320 


92 


93 


.57782 


1.45465 


.23930 


.58390 


93 


94 


.40763 


1.04702 


.16723 


.41667 


94 


95 


.29116 


.75586 


.11832 


.29835 


95 


96 


.21260 


.54326 


.08558 


.21277 


96 


97 


.15846 


.38480 


.06318 


.14959 


97 


98 


.11738 


.26742 


.04636 


.10323 


98 


99 


.08783 


.17959 


.03437 


.06886 


99 


100 


.06844 


.11115 


.02653 


.04233 


100 


101 


.05070 


.06045 


.01946 


.02287 


101 


102 


.03445" 


.02596 


.01312 


.00975 


102 


103 


.01970 


.00626 


.00742 


.00233 


103 


104 


.00626 


| .00000 


.00233 


.00000 


| 104 



112 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XIV. 
Commutation Columns for Annuities. 
Northampton Experience. One Life. 



X 


O *j 


a *i 


am 


o w 


X 


> 


CD +2 






§E- 


> 


(TC 


r+ © 


r+ © 


n- ^ 


^ ^d 


en? 


CD 


^ 


hd 


CD 


CD 








CD 


>-i 


n 






Dx 


Nx 


Dx 


Nx 




1 


8238.094 


95264.271 


8160.380 


82473.4470 


1 


2 


6605.891 


88658.380 


6481.845 


75991.602 


2 


3 


5857.682 


82800.698 


5693.453 


70298.149 


3 


4 


5303.130 


77497.568 


5105.833 


65192.316 


4 


5 


4896.289 


72601.279 


4669.626 


60522.690 


5 


6 


4526.005 


68075.274 


4275.782 


56246.908 


6 


7 


4210.787 


63864.487 


3940.464 


52306.444 


7 


8 


3935.921 


59928.566 


3648.403 


48658.041 


8 


9 


3696.816 


56231.750 


3394.538 


45263.503 


9 


10 


3483.958 


52747.792 


3168.891 


42094.612 


10 


11 


3287.423 


49460.369 


2962.133 


39132.479 


11 


12 


3103.254 


46357.115 


2769.609 


36362.870 


12 


13 


2928.964 


43428.151 


2589.398 


33773.472 


13 


14 


2764.236 


40663.915 


2417.705 


31355.767 


14 


15 


2608.555 


38055.360 


2262.828 


29092.939 


15 


16 


2461.433 


35593.927 


2115.061 


26977.878 


16 


17 


2321.099 


33272.828 


1975.659 


25002.219 


17 


18 


2186.469 


31086.359 


1843.589 


23158.630 


18 


19 


2057.425 


29028.934 


1718.337 


21440.293 


19 


20 


1934.200 


27094.734 


1600.182 


19840.111 


20 


21 


1816.249 


25278.485 


1488.426 


18351.685 


21 


22 


1704.122 


23574.363 


1383.363 


16968.322 


22 


23 


1598.555 


21975.807 


1285.424 


15682.898 


23 


24 


1499.178 


20476.629 


1194.141 


14488.757 


24 


25 


1405.642 


19070.981 


1009.074 


13379.683 


25 


26 


1318.613 


17752.374 


1029.809 


12349.874 


26 


27 


1234.781 


16517.593 


955.9648 


11393.909- 


27 


28 


1156.849 


15360.744 


887.1829 


10506.726 


28 


29 


1083.541 


14277.203 


823.1231 


9683.603 


29 


30 


1014.590 


13262.6132 


763.4728 


8920.130 


30 


31 


949.7496 


12312.8636 


707.9384 


8212.192 


31 


32 


888.7835 


11424.0797 


656.2445 


7555.947 


32 


33 


831.4696 


10592.6196 


608.1354 


6947.812 


33 


34 


777.5995 


9815.0106 


563.3687 


6384.443 


34 


35 


726.9740 


9088.0366 


521.7219 


5862.721 


35 


36 


679.4069 


8408.6297 


482.9850 


5379.736 


36 


37 


634.7215 


7773.9082 


446.9616 


4932.774 


37 


38 


592.7512 


7181.1570 


413.4690 


4519.305 


38 


39 


553.3388 


6627.8182 


382.3360 


4136.969 


39 


40 


516.3360 


6111.4822 


353.4028 


3783.566 


40 


41 


481.4672 


5630.0150 


326.4280 


3457.138 


41 


42 


448.6196 


5181.3954 


301.2884 


3155.850 


42 


43 


417.6858 


4763.7096 


277.8672 


2877.983 


43 


44 


388.6907 


4375.0189 


256.1322 


2621.851 


44 


45 


361.4911 


4013.5278 


235.9674 


2385.884 


45 


46 


336.0095 


3677.5183 


217.2648 


2168.619 


46 


47 


312.1350 


3365.3833 


199.9235 


1968.695 


47 


48 


289.7723 


3075.6110 


183.8491 


1784.8455 


48 



SINGLE LIFE COLUMNS. 



113 



TABLE NO. XIV. (Concluded. 
Commutation Columns for Annuities. 
Northampton Experience. One Life. 



X 

> 

o 


9a 

<r* CD 

CD 


<T> 
-i 


Six Per 
Cent 


Six Per 
Cent 


X 

> 

CD 




Dx 


Nx 


Dx 


Nx 




49 


268.8317 


2806.7793 


168.9541 


1615.8914 


49 


50 


249.1239 


2557.6554 


155.1018 


1460.7896 


50 


51 


230.5501 


2327.1053 


142.1741 


1318.6155 


51 


52 


213.0855 


2114.0198 


130.1643 


1188.4512 


52 


53 


196.7616 


1917.2582 


119.0590 


1069.3922 


53 


54 


181.5096 


1735.7492 


109.0445 


960.3477 


54 


55 


167.2630 


1568.4862 


99.3090 


861.0387 


55 


56 


153.9621 


1314.5224 


90.5495 


770.4892 


56 


57 


141.5486 


1272.9755 


82.4637 


688.0257 


57 


58 


129.9684 


1143.0071 


75.0027 


613.0230 


58 


59 


119.1701 


1923.8370 


68.1225 


544.9005 


59 


60 


109.1053 


914.7317 


61.7806 


483.1199 


60 


61 


99.7290 


815.0027 


55.9385 


427.1814 


61 


62 


90.0983 


724.9044 


50.5599 


376.6215 


62 


63 


82.9191 


641.9853 


45.6363 


230.9852 


63 


64 


75.4030 


566.5823 


41.1081 


289.8770 


64 


65 


68.4567 


498.1256 


36.9690 


252.9079 


65 


66 


62.0009 


436.1247 


33.1669 


219.7410 


66 


67 


56.0047 


380.1200 


29.6766 


190.0644 


67 


68 


50.4390 


329.6810 


26.4752 


163.5891 


68 


69 


45.2764 


284.4046 


23.5412 


140.0479 


69 


70 


40.4911 


243.9135 


20.8545 


119.1934 


70 


71 


36.0589 


207.8546 


18.3965 


100.7969 


71 


72 


31.9569 


175.8977 


16.1500 


84.6469 


72 


73 


28.1639 


147.7338 


14.0988 


70.5481 


73 


74 


24.6596 


123.0742 


12.2281 


58.3199 


74 


75 


21.4253 


101.6489 


10.5241 


47.7958 


75 


76 


18.4430 


83.1059 


8.9737 


38.8221 


76 


77 


15.7762 


67.4297 


7.5989 


31.2232 


77 


78 


13.3916 


54.0381 


6.3935 


24.8297 


78 


79 


11.3132 


42.7249 


5.3503 


19.4794 


79 


80 


9.4630 


33.2619 


.4.4330 


15.0463 


80 


81 


7.8017 


25.4602 


3.6204 


11.4259 


81 


82 


6.3322 


19.1280 


2.9107 


8.5152 


82 


83 


5.0371 


14.0909 


2.2936 


6.2216 


83 


84 


3.8843 


10.2065 


1.7520 


4.4697 


84 


85 


2.9405 


7.2660 


1.3138 


3.1559 


85 


86 


2.1832 


5.0829 


.9661 


2.1897 


86 


87 


1.5917 


3.4912 


.6977 


1.4919 


1 87 



1.1335 



.4922 



89 


.8062 


1.5515 


90 


.5698 


.9817 


91 


.4011 


.5806 


92 


.2696 


.3109 


93 


.1712 


.1398 


94 


.0917 


.0481 


95 


.0388 


.0092 


96 


.0092 


.0000 



.3469 
.2428 
.1693 
.1127 
.0709 
.0376 
.0158 
.0037 



.9997 



.6528 
.4100 
.2407 
.1280 
.0571 
.0195 
.0037 
.0000 



89 
90 
91 
92 

I 93 
94 
95 

I 96 



114 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XV. 

Commutation Columns for Annuities. 

American Experience. One Life. 



X 

> 

OK} 
O 


Five Per 
Cent 


05 ;-;■ 




am 

05 £• 
05 


> 

05 




Dx 


Nx 


Dx 


Xx 




10 


61391.48 


1013248.23 


55839.45 


798121.48 


10 


11 


58030.00 


955218.23 


52284.19 


745837729 


11 


12 


54851.27 


900366.96 


48953.98 


696883.31 


12 


13 


51845.27 


848521.69 


45834.73 


651048.58 


13 


14 


49002.70 


799518.99 


42912.92 


608135.66 


14 


15 


46314.72 


753204.27 


40176.37 


567959.29 


15 


16 


43772.56 


709431.71 


37612.91 


530346.38 


16 


17 


41368.79 


668062.92 


35212.03 


495134.35 


17 


18 


39095.93 


628966.99 


32963.50 


462170.85 


18 


19 


36946.51 


592020.48 


30857.35 


431313.50 


19 


20 


34913.91 


557106.57 


28884.66 


402428.84 


20 


21 


32991.93 


524114.74 


27037.00 


375391.84 


21 


22 


31174.00 


492940.74 


25306.24 


350085.60 


22 


23 


29454.76 


463485.98 


23685.01 


326400.59 


23 


24 


27828.90 


435657.08 


22166.57 


304234.02 


24 


25 


26291.45 


409365.63 


20744.34 


283489.68 


25 


26 


24837.50 


384528.13 


19412.30 


-264077.38 


26 


27 


23462.45 


361065.68 


18164.60 


245912.78 


27 


28 


22162.02 


338903.66 


16995.96 


228916.82 


28 


29 


20932.25 


317971.41 


15901.41 


213015.41 


29 


30 


19769.12 


298202.29 


14876.14 


198139.27 


30 


31 


18669.08 


279533.21 


13915.84 


184223.43 


31 


32 


17628.76 


261904.45 


13016.43 


171207.00 


32 


33 


16644.74 


245259.71 


12173.95 


159033.05 


33 


34 


15714.00 


229545.71 


11384.73 


147648.32 


34 


35 


14833.53 


214712.18 


10645.48 


137002.84 


35 


36 


14000.78 


200711.40 


9953.05 


127049.79 


36 


37 


13212.90 


187498.50 


9304.329 


117745.461 


37 


38 


12467.51 


175030.99 


8696.614 


109048.847 


38 


39 


11762.08 


163268.91 


8127.166 


100921.681 


39 


40 


11094.62 


152174.29 


7593.637 


93328.044 


40 


41 


10462.81 


141711.483 


7093.643 


86234.401 


41 


42 


9864.861 


131846.622 


6225.142 


79609.259 


42 


43 


9298.785 


122547.837 


6186.057 


73423.202 


43 


44 


8762.846 


113784.991 


5774.526 


67648.676 


44 


45 


8255.196 


105529.795 


5388.673 


62260.003 


45 


46 


7774.327 


97755.468 


5021.127 


57238.887 


46 


47 


7318.514 


90436.954 


4687.533 


52551.349 


47 


48 


6886.373 


83550.581 


4369.132 


48182.217 


48 


49 


6476.407 


77074.174 


4070.262 


44111.955 


49 


50 


6087.169 


70987.005 


3789.544 


40322.411 


50 


. 51 


5717.409 


65269.596 


3525.773 


36796.638 


52 


52 


5365.974 


59903.622 


3277.836 


33518.802 


52 



SINGLE LIFE COLUMNS. 



115 



TABLE NO. XV. (Concluded.) 

Commutation Columns for Annuities. 

American Experience. One Life. 



> 

era 

CD 


2 < 

£- CD 

CD 
-i 


CD 5" 
CD 


Six Per 
Cent 


QC/2 
CD S" 

P * 

■* hd 

CD 


* 

> 
era 

CD 




Dx 


Nx 


Dx 

rs 


Xx 




53 


5031.809 


54871.813 


3444.712 


30474.090 


53 


54 


4713.926 


50157.887 


2825.455 


27648.635 


54 


55 


4411.358 


45746.529 


2619.154 


25029.481 


55 


56 


4123.270 


41623.259 


2425.014 


22604.467 


56 


57 


3848.837 


37774.422 


2242.256 


20362.211 


" 57 


58 


3587.354 


34187.068 


I 2070.205 


18292.006 


58 


59 


3338.167 


30848.901 


| 1908.230 


16383.776 


59 


60 


3100.617 


27748.284 


| 1755.717 


14628.059 


60 


61 


2874.143 


24874.141 


I 1612.122 


13015.937 


61 


62 


2658.227 


22215.914 


| 1471.947 


11538.990 


62 


63 


2452.424 


19763.490 


| 1349.746 


10189.244 


63 


64 


2256.365 


17507.125 


| 1230.124 


8959.120 


64 


65 


2069.681 


15437.444 


I 1117.702 


7841.418 


65 


66 


1892.026 


13545.418 


| 1012.124 


6829.2939 


66 


67 


1723.173 


11822.245 


913.1015 


5916.1924 


67 


68 


1562.922 


10259.323 


820.3721 


5095.8203 


68 


69 


1411.092 


8848.231 


733.6896 


4362.1307 


69 


70 


1267.615 


7580.616 


652.8718 


3709.2589 


70 


71 


1132.411 


6448.205 


577.7342 


3131.5247 


71 


72 


1005.511 


5442.6945 


508.1526 


2623.3721 


72 


73 


887.021 


4555.6735 


444.0423 


2179.3298 


73 


74 


777.0489 


3778.6246 


385.3208 


1794.0090 


74 


75 


675.6422 


3102.9824 


331.8748 


1462.1342 


75 


76 


582.7441 


2520.2373 


283.5431 


1178.5911 


76 


77 


498.2129 


2022.0244 


240.1262 


938.4649 


77 


78 


421.7896 


1600.2348 


I 201.3742 


737.0907 


78 


79 


353.1676 


1247.0672 


167.0214 


570.0693 


79 


80 


292.0417 


955.0255 


| 136.8104 


433.2589 


80 


81 


237.9538 


717.0717 


110.4208 


322.83808 


81 


82 


190.6792 


526.3925 


87.64858 


235.18950 


82 


83 


149.9471 


376.4454 


68.27522 


166.91428 


83 


84 


115.4506 


260.9948 


52.07200 


114.84228 


84 


85 


86.7134 


174.28137 


38.74165 


76.10063 


85 


86 


63.13136 


111.15001 


27.93640 


48.16099 


86 


87 


44.15102 


66.99899 


19.35527 


28.80572 


| 87 


88 


29.30700 


37.69199 


1 12.72664 


16.07908 


88 


89 


18.23477 


19.45722 


7.84380 


8.23528 


89 


90 


10.49171 


8.96551 


1 4.47.049 


3.76479 


90 


91 


5.45024 


3.51527 


2.30043 


1.46436 


91 


92 


2.42682 


1.08844 


1.01464 


.44972 


92 


93 


.84532 


.24312 


.35009 


.09963 


93 


94 


.21401 


| , .02912 


.08779 


.01183 


94 


95 


.02912 


.00000 


.01183 


.00000 


95 



116 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XVI. 

Carlisle Table, Makeham's Formula. 

Annuity Commutation Columns. 



X 

> 

CTQ 

CD 


Two Lives 
Equal Ages 
Five Per 
Cent 


Two Lives 
Equal Ages 
Five Per 
Cent 


X 

> 
ere 

CD 


Three Lives 
Equal Ages 
Five Per 
Cent 


Three Lives 
Equal Ages 
Five Per 
Cent 




Dxx 


Nxx 




Dxxx 


Nxxx 


10 


2561958 1 


37854571 


10 


1655025 


22124056 


11 


2415546 


35439025 


11 


1552616 


20571440 


12 


2277030 


33161995 


12 


1456090 


19115350 


13 


2145939 


31016056 


13 


1365074 


17750276 


14 


2021770 


28994286 


14 


1279154 


16471122 


15 


1904240 


27090046 


15 


1198135 


15272987 


16 


1792878 


25297168 


16 


1121622 


14151365 


17 


1687426 


23609748 


17 


1049407 


131019579 


18 


1587487 


22022261 


18 


9812252 


121207327 


19 


1492728 


20529533 


19 


9167883 


112039444 


20 


1406003 


19126530 


20 


8560157 


103479287 


21 


1318035 


17808495 


21 


7986898 


95492389 


22 


1237681 


16570814 


22 


7444571 


88047818 


23 


1160987 


15409827 


23 


6932952 


81114866 


24 


1088476 


14321351 


24 


6449110 


74665756 


25 


1018388 


13302963 


25 


5980486 


68685270 


26 


9526080 


123503550 


26 


5544306 


63140964 


27 


8909353 


114594197 


27 


5138682 


58002282 


28 


8332344 


106261853 


28 


4761798 


53240484 


29 


7789530 


98472323 


29 


4410713 


48829771 


30 


7281153 


91191170 


30 


4084509 


44745262 


31 


6804023 


84387147 


31 


3780792 


40964470 


32 | 6356768 


78030379 


1 32 


3498510 


| 37465960 


53 | 5937148 


72093231 


[ 33 


3235864 


| 34230096 


34 | 5543553 


66549678 


34 


2991579 


31238517 


35 | 5174056 


61375622 


35 


2764135 


28474382 


36 4S27451 


56548171 


1 36 


2552616 


25921766 


37 ! 4502264 


52045907 


1 37 


2355854 


23565912 


38 1 43 97070 


47848837 


1 38 


2172781 


21393131 


39 


3910741 


43938096 


| 39 


2002564 


| 19390567 


40 


3641913 


40296183 


1 40 


1844083 


1 17546484 


41 


3389493 


36906690 


1 41 


1696607 


15849877 


42 


3152430 


33754260 


1 42 


1559350 


14290527 


43 


2930041 


30824219 


1 43 


1431795 


12858733 


44 


2721168 


28103051 


1 44 


I 1313097 


1 11545636 


45 


2524998 


25578053 


1 45 


1202682 


10342954 


46 


2340869 


23237184 


| 46 


1100068 


92428860 


47 


2167877 


21069307 


| 47 


1004613 


82382734 


48 


2005546 


19063761 


I 48 


9158931 


73223803 


49 


1853020 


17210741 


| 49 


8336002 


64887801 


50 


1709820 


15500921 


| 50 


7571081 


57316720 


51 


1575368 


13925553 


1 51 


6861200 


50455520 


52 


1449257 


12476296 


1 52 


6203544 


44251976 


53 


1330845 


11145451 


1 53 


5593808 


38658168 


54 


1219741 


99257100 


I 54 


5029357 


33628811 


55 


1115641 


88100691 


| 55 


4508072 


291207~99 


56 


1018024 


77920452 


|. 56 


4026590 


25094209 


57 


1 9266926 


68653526 


1 57 


3583428 


21510781 



JOINT LIVES — EQUAL AGES. 



117 



TABLE XO. XVI. (Concluded) 

Carlisle Table, Makeham's Formula, 
Annuity Commutation Columns. 



;* 


C^EH 


C^HH 


X 


Q^ M H 


CE E H 


OP 


2<£ 3 




> 
ore? 


lire 
Iqua 
ive 
ent 


lire 
qua 
ive 
ent 


re 


^z-r 


^~r 


CD 


TJS- CD 


'■g ~ o 




to K. 1' 


*>1 






5 >V. 




cn % 


<H % 




Oq < 


in < 




o> Ji 






CD cd 


CP CD 




CO 


00 




CO CO 


CO CO 




Dxx 


Xxx 


5S~ 


Dxxx 


Xxxx 


58 (8411580 


60241946 


3175456 


18335325 


59 17612907 


52629039 


59 


2801626 


15533699 


60 J6867078 


45161961 


60 


2459444 


13074255 


61 16172150 


39589811 


61 


2147476 


10926779 


62 


5525524 


34064287 


62 


1863921 


9062858 


63 


4925399 


29138SS8 


63 


1607404 | 


7455454 


64 


4369970 


2476891S 


64 


1376497 


6078957 


65 


3857167 


20911751 


65 


1169647 


4909310 


66 


33S5249 


17526502 


66 | 9854458 


39238642 


67 2952683 


14573819 
12015758 


67 


8225583 


31013059 


68 J2558061 


68 


6796769 


24216290 


69 


2199847 


9815911 


69 


5554173 


18662117 


70 


1876408 


7939503 


70 


4483486 


14178631 


71 


1586449 


6353054 


71 


3571576 


1060755 


72 


1328203 


5024851 


72 


2803572 


7803483 


73 


1100262 


39245894 


73 


2165976 


5637507 


74 


9009650 


30236244 


74 


1644621 


3992886 


75 


7283686 


22952558 


75 


1224970 


27679162 


76 


5806591 


17145967 


76 


8934602 


18744560 


77 


4559099 


12586868 


77 


6369517 


12375043 


78 | 3519872 


9066996 


78 


4427647 


~ 7947396 


79 2668951 


6398045 


79 


2995632 


4951764 


80 1 1983143 


4414902 


80 


1966088 


2985676 


81 1441794 


2973108 


81 


1248882 


17367942 


82 1 1023407 


1949700.9 


82 


7653038 


9714904 


83 


707462.5 


1242238.4 


83 


4507244 


5207660 


84 


475301.2 


766937.2 


84 


2543337 


2664323 


85 


309413.9 


457523.3 


85 


1368847 


12954856 


86 


194589.0 


262934.3 


86 


6995471 


5959385 


87 


117865.4 


145068.79 


87 


3379202 


2580133 


88 1 68523.19 


96545.60 


88 


1534919 


10452637 


89 1 38076.09 


38469.51 


89 


6514819 


39378180 


90 | 20136.47 


18333.049 


90 1 2567401 


13704174 


91 ! 10093.86 


8239.189 


91 


9336817 


4367357 


92 4761.532 


3477.657 


92 


3099757 


1267599.6 


93 1 2109.411 


1368.2463 


93 


936578.5 


331021.1 


94 ' 868.9034 


499.3429 


94 


253714.0 


77307.14 


95 * 331.7701 


167.5728 


95 I 61451.39 


15855.75 


96 ! 116.0245 


51.5483 


96 


12986.18 


2869.567 


97 i 37.1483 


14.4000 
3.6801 


97 


2417.569 


451.9977 


98 10.7199 


98 


391.1616 


60.8361 


99 | 2.8976 


.7825 


99 


54.7671 


6.0689 


100 : .6325 


.1500 


100 


5.5437 


.5253 


101 I .1159 


.0342 


101 


.4635 


.0618 


102 ! .0276 


.0066 


102 


.0552 


.0066 


103 


| .0066 


.0000 


1 103 


.0066 


.0000 



118 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XVII. 

Carlisle Table, Makeham's Formula, 
Annuity Commutation Columns. 



x 


OgHH 


Q^HH 


X 


Cim H H 


Three L 
Equal A 
Six Per 
Cent 


en* 


wo Live 
qual A 
x Per 
ent 


wo Liv< 
qual A 
ive Per 

ent 


> 

CTQ 


hree Li 
qual A 
lix Per 

ent 




3* 

Ul 


Ul 




crq < 

Ul Ul 


crq 5 - 

% % 

50 Ul 




Dxx 


Nsx 




Dxxx 


Nxxx 


10 


2330270 | 


30324753 I 


10 


1550355 


17926548 


11 


2176372 | 


28148391 1 


11 


1398885 


16527663 


12 


2032217 


26116164 | 


12 


1299542 


15228121 


13 


1897152 | 


24219012 | 


13 


1206816 


14021305 


14 


1770516 


22448496 | 


14 


1120188 


12901117 


15 


1651869 | 


20796627 | 


15 


1039340 


118617765 


16 


1540586 


19256041 


16 


9637749 


108980016 


17 


1436288 


17819753 1 


17 


8932276 


100047740 


18 


1338479 


16481274 


18 


8273181 


91774559 


19 


1246711 


15234563 | 


19 


7656923 


84117636 


20 


1160720 


14073843 


20 


7081900 


77035736 


21 


1080137 


12993706 


21 


6545308 


70490428 


22 


1004487 


119892185 


22 


6043297 


64447131 


23 


9335691 


110556494 


23 


5574901 


58872230 ■ 


24 


8670044 


101886450 


24 


5136913 


53735317 


25 


8035250 


93851200 


25 


4718700 


49016617 


26 


7445493 


86405707 


26 


4333277 


44683340 


27 


6897898 


79507809 


27 


3978363 


40704977 


28 


6390037 


73117772 


28 


3651802 


37053175 


29 


5917400 


67200372 


29 


3350668 


33702507 


30 


5479037 


61721335 


30 


3073568 


30628939 


31 


5071684 


56649651 


31 


2818183 


27810756 


32 


4693600 


51956051 


32 


2583170 


25227586 


33 


4342415 


47613636 


33 


2366703 


22860883 


34 


4016290 


43597346 


34 


2167387 


20693496 


35 


3713226 


39884120 


35 


1983716 


18709780 


36 


3431804 


36452316 


36 


1814636 


16895144 


37 


3170428 


33281888 


37 


1658958 


15236186 


38 


2927634 


30354254 


38 


1515621 


13720565 


39 


2702173 


27652081 


39 


1383675 


12336890 


40 


2492683 


25159398 


40 


1262170 


11074720 


41 


2298024 


22861374 


41 


1150276 


99244440 


42 


2117140 


20744234 


42 


1047243 


88772006 


43 


1949223 


18795011 


43 


9525071 


79246935 


44 


1793191 


17001820 


44 


8653042 


70593893 


45 


1648221 


15363599 


45 


7850650 


62743243 


46 


1513614 


13839985 


46 


7113075 


55630168 


47 


1388529 


12451456 


47 


6434584 


49195584 


48 


1272440 


11179016 


48 


5811617 


43383967 


49 


1164577 


10014439 


49 


5238971 


38144996 


50 


1064441 


89499984 


50 


4713348 


33431648 


51 


9714873 


79785111 


51 


4231118 


29200530 


52 


8852869 


70932242 


52 


3789471 


25411059 


53 


8052846 


62879396 


53 


3384772 


22026287 


54 


7310938 


55568458 


54 


3014519 


19011768 


55 


6623876 


48994582 


55 


2676515 


16335253 


56 


5987791 


42956791 


56 


2368153 


13967100 


67 


5398727 


37558064 


57 


2087634 


11879466 



JOINT LIVES — EQUAL AGES. 



119 



TABLE NO. XTII. (Concluded) 

Carlisle Table — Makekam's Formula. 

Annuity Commutation Columns. 



* 


QffiEH 


QCfl K H 


! * 


QCQeM 


OCCHH 


> 


i**i 




> 


3 * £ *< 


3 x e -s 


W 


rt idP° 


QTQ 


<r- hjjp CD 


<rf hri o: cd 


CD 


CD —r 


CD ~ f 


CD 


o ~° 


CD —^ 




"►? 


">$ 




*>z 


*>£ 




en » 


W g> 




OK? < 


crq o 




CD M 


CD M 




CP CD 


CD CD 




m 


CD 




'A CO 


03 oa 




Dxx 


Nxx 




Dxxx 


Nxxx 


58 


4854190 


32703874 


| 58 


1832555 


10046911 


59 


4351843 


28352031 


| 59 


1601522 


8445389 


60 


3888467 


24463064 


i 60 


1319554 


7052835 


61 


3461998 


21001566 


1 61 


1204531 


5848304 


62 


3070056 


17931510 


i 62 


1035622 


48126817 


63 


2710801 


15220709 


1 63 


8846700 


39280117 


64 


2382418 


12838291 


J 64 


7504381 


31775736 


65 


2083012 


10755279 


| 65 


6316526 


25459210 


66 


1810911 


8944368 


| 66 


5271563 


20187647 


67 


1564613 


7379755 


i 67 


4358698 


15828949 


68 


1342717 


60370380 


1 68 


3567598 


12261351 


69 


1143798 


48932404 


69 


2887861 


9372490 


70 


9664231 


39268173 


70 


2309172 


7064318 


71 


8093759 


31174414 


71 


1822148 


5242170 


72 


6712309 


24462105 


72 


1416834 


3825336 


73 


5507909 


18954196 


73 


1084285 


27410513 


74 


4467679 


14486517 


74 


8155302 


19255211 


75 


3577739 


10908778 


75 


6017042 


13238169 


76 


3825286 


8083492 


76 


4347268 


8890901 


77 


2197372 


5886120 


77 


3069948 


5820953 


78 


1680472 


4205648 


78 


2113882 


3707071 


79 


1262211 


2943437.5 


79 


1416706 


22903645 


80 


929026.9 


2014410.6 


80 


9210374 


13693271 


81 


669054.5 


1345356.1 


81 


5795351 


7897920 


82 


470424.4 


874931.7 


82 


3517.835 


4380085 


83 


322128.0 


552803.7 


83 


2052277 


23278084 


84 


214376.5 


338427.2 


84 


1147126 


11806824 


85 


138239.3 


200187.85 


85 


6115703 


5691121 


86 


86117.96 


114069.89 


86 


3095940 


2595181 


87 


51670.78 


62399.11 


87 


1481401 


11137798 


88 


29756.37 


32642.74 


88 


6665428 


4472370 


89 


16378.66 


16264.08 


89 


2802389 


1669981 


90 


8580.100 


7683.98 


90 


1093963 


5760176 


91 


4260.396 


3423.584 


91 


3940866. 


1819310. 


92 


1990.776 


1432.8083 


92 


1295995 


523314.9 


93 


873.6162 


559.1921 


93 


387885.6 


135429.3 


94 


356.4633 


202.72881 


94 


104087.3 


31341.98 


95 


134.9851 


67.7437! 


95 


24972.24 


6369.736 


96 


46.6738 


21.0699 


96 


5227.470 


1142.2656 


97 


14.8306 


6.2393 


97 


963.9864 


178.2792 


98 


4.8154 


1.4240] 


98 


154.5013 


23.7779 


99 


1.1278 


.29621 


99 


21.4279 


2.3499 


100 


.2387 


.05751 


100 


2.1485 


.2014 


101 


.0445 


.013CU 


101 


.1779 


.0235 


102 


.0105 


.00251 


102 


.0210 


.0025 


103 


.0025 


.0000! 


103 


.0025 


.0000 



120 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XVIII. 

American Experience Table, Makeham's Formula, 

Annuity Commutation Columns 



X 

> 

crq 

o 


Two Lives 
Equal Ages 
Three & One 
Half Per 
Cent 


Two Lives 
Equal Ages 
Three & One 
Half Per 
Cent 


> 
crq 


Three Lives 
Squal Ages 
Three & One 
Half Per 

Cent 


Three Lives 
Equal Ages 
Three & One 
Half Per 
Cent 




Dxx 


Nxx 


*■ 


Dxxx 


Mxxx 


10 


7100669 


128026372 


10 


7188715 * 


111888^95 


11 


6755941 


121270431 


11 


6709661 


105178534 


12 


6427697 


114842734 


12 


6334687 


98843847 


13 


6115297 


108727437 


13 


5980^16 


92863331 


14 


5817984 


102909453 


14 


5646004 


87217327 


15 


5534918 


97374535 


15 


5329904 


81887423 


16 


5265432 


92109103 


16 


5031226 


76856197 


17 


5008879 


87100224 


17 


4749019 


72107178 


18 


4764653 


82335571 


18 


4482395 


67624783 


19 


4532074 


77803497 


19 


4230374 


63394409 


20 


4310684 


73492813 


20 


3992296 


59402113 


21 


4099871 


693-92942 


21 


3767290 


55634823 


22 


3899137 


65493805 


22 


3554648 


52080175 


23 


3707930 


61785875 


23 


3353600 


48726575 


24 


3525885 


58259990 


24 


3163644 


45562931 


25 


3352500 


54907490 


25 


2984060 


42578871 


26 


3187302 


51720188 


26 


2814227 


38764644 


27 


3029914 


48690274 


27 


2653630 


37111014 


28 


2879982 


45810292 


28 


2501783 


34609231 


29 


2737164 


43073128 


29 


2358231 


32251000 


30 


2600952 


40472176 


30 


2222305 


30028695 


31 


2471172 


38001004 


31 


2093775 


27934920 


32 


2347479 


35653525 


32 


1972190 


25962730 


33 


2229494 


33424031 


33 


1857057 


24105673 


34 


2117021 


31307010 


34 


1748131 


22357542 


35 


2009672 


29297338 


35 


1644916 


20712626 


36 


1907282 


27390056 


36 


1547203 


19165423 


37 


1809584 


25580472 


37 


1454667 


17710756 


38 


1716306 


23864166 


38 


1366969 


16343787 


39 


1627183 


22236983 


39 


1283785 


15060002 


40 


1542093 


20694890 


40 


1204212 


13855790 


41 


1460762 


19234128 


41 


1130178 


12725612 


42 


1383051 


17851077 


42 


1083935 


116663471 


43 


1308683 


16542394 


43 


9919034 


106744437 


44 


1237580 


15304814 


44 


9279996 


97464441 


45 


1169500 


14135314 


45 


8672778 


88791663 


46 


1104287 


13031027 


46 


8095641 


80696022 


47 


1041829 


119891983 


47 


7547321 


73148701 


48 


9819618 


110072365 


48 


7026035 


66122666 


49 


9245145 


100827220 


49 


6529939 


59592727 


50 


8694350 


92132870 


50 


6058470 


53534257 


51 


8165466 


83967404 


51 


5609839 


47924418 


52 


7657489 


76309915 


52 


5182974 


42741444 


53 


7169272 


69140643 


53 


4776743 


37964701 


54 


6700205 


62440438 


54 


4390577 


33574124 



JOINT LIVES— EQUAL AGES. 



121 



TABLE NO. XXIII. (Concuded) 
American Experience Table, Makeham's Formula, 
Annuity Commutation Columns. 



X 


O^HKH 


O3HEH 


X 


OlMHH 


OSHHH 


> 


« p cr.a 3 


2 p c 6 3 


> 


cd p s-,a cr 


2 p tf& cr 


B E- 1 £ O 


3 — ~s £ 


3 -1 c: 4 


P *-* | -s S3 ►I 


Bq 


~">CD p ° 


^ rt « p ° 


an? 


^^ ffi M (C 


<rt * •"& CD Q3 (D 


CD 


^o &r 


hd^ ^r 


cd 


^a> — cd 


hg CD •— CD 






" *°£c? 




* &>£ 


5 fc>£ 




o<* £ 


o<* S 




ow < 


OC£ <j 




a ^ 
o m 


CD 




a cp cd 

g m ui 


g CO w 




Dxx 


Nxx 




Dxxx 


Nxxx 


55 


6249185 


56191253 


55 


4023411 


29550713 


6 


5815617 


50375636 


6 


3674714 


25875999 


7 


5398665 


44976971 


7 


3343720 


22532279 


8 


4997784 


39979187 


8 


3029957 


19502322 


9 


4612537 


35366650 


9 


2733067 


16769255 


60 


4242434 


31124216 


60 


2452636 


14316619 


1 


3887366 


27236850 


1 


2188587 


12128032 


2 


3546890 


23689960 


2 


1940539 


10187493 


3 


3221295 


20468665 


3 


1708704 


8478789 


4 


2910385 


17558280 


4 


1492854 


6985935 


5 


2614375 


14943905 


5 


1293044 


5692891 


6 


2333362 


12610543 


6 


1109186 


45837046 


7 


2067834 


10542709 


7 


9414020 


36423026 


8 


1818091 


8724618 


8 


7895791 


28527235 


9 11584450 


7140168 


9 


6535236 


21991999 


70 |1367422 


5772746 


70 


5330499 


16661500 


1 11167330 


46054159 


1 


4277330 


12384170 


2 | 9845444 


36208715 


2 


3370581 


9013589 


3 8193524 


28015191 


3 


2603328 


6410261 


4 1 6717844 


21297347 


4 


1966246 


4444015 


5 | 5417067 


15880280 


5 


1448470 


2995545 


6 | 4288522 


11591758 


6 


1037994 


19575515 


7 | 3326534 


8265224 


7 


7214255 


12361260 


8 1 2522044 


5743180 


8 


4845100 


7516160 


9 | 1864614 


3878566 


9 


3133487 


4382673 


80 i 1340320 


25382455 


80 


1942793 


2439880 


1 1 9338504 


16043951 


1 


1149474 


12904058 


2 1 6284720 


9759231 


2 


6456294 


6447764 


.3 | 4069518 


5689713 


3 


3422465 


3025299 


4 | 2524653 


3165060 
1671060.9 


4 


1701364 


13239350 


5 I 1493999 


5 


7879353 


5359997 


6 1 838235.3 


832825.6 


6 


3368868 


19911290 


7 | 443477.4 


389348.2 


7 


1318904 


6722249 


8 I 219790.3 


169557.9 


8 


4681533 


2040716 


9 | 101282.4 


68275.47 


9 


1489865 


5508511 


90 | 43079.25 


25196.22 


90 


4204534 


1303977 


1 1 16742.08 


8454.137 


1 


1036334 


267643.4 


2 I 5905.151 I 


2548.986 


2 


220852.6 


46790.85 


3 ! 1867.992 ! 


680.994 


3 


39975.03 


6815.821 


4 | 521.1980! 


159.7956 


4 


5993.776 


822.0448 


5 1 128.0922: 


31.7034 


5 


744.6475 


77.3973 


6 


26.8197! 


4.8837 


6 


72.4132 


4.9841 


7 


4.3010! 


.5827 


7 


4.7311 


.2530 


8 


.5495! 


.0332 


8 


.2198 


.0332 


9 


.0332| 


.0000 


99 


.0332 


.0000 



122 



FINANCE AND LIFE INSURANCE, 



TABLE NO. XIX. 

American Experience Table, Makeham's Formula, 

Annuity Commutation Columns. 



X 


O^MH 


O^HH 


X 


Q^IHH 


Q1HH 


> 


s.» g P 


8 <% * 


> 
cm 


2 3"»° ^ 

c+ ® g CD 


2 5'*° &* 

2- CD £ 2 

^* ™ OD CD 


CD 


►u~r 


hd^-r 


CD 


hd^tt 


hd*-® 




s^Sf 


*>% 










crq ® 


cm g 




cm < 


Cm <! 




cp « 


£ M 




CD CD 


CD CD 




02 




Ul 01 


02 ca 




Dxx 


Nxx 




Dxxx 


Nxxx 


10 


6149080 


88895620 


10 


6154061 


79411583 


11 


5766951 


83128669 


11 


5727361 


73684222 


12 


5408390 


77720279 


12 


5330130 


68354092 


13 


5072024 


72648255 


13 


4960237 


63393855 


14 


4756497 


67891758 


14 


4615895 


58777960 


15 


4460432 


63431326 


15 


4295217 


54482743 


16 


4182643 


59248683 


16 


3996599 


50486144 


17 


3922009 


55326674 


17 


3718535 


46767609 


18 


3677479 


51649195 


18 


3459625 


43307984 


19 


3447998 


48201197 


19 


3218464 


40089520 


20 


3232721 


44968476 


20 


2993945 


37095575 


21 


3030697 


41937779 


21 


2784846 


34310729 


22 


2843133 


39096646 


22 


2590119 


31720610 


23 


2663212 


36433434 


23 


2408716 


29311894 


24 


2496280 


33937154 


24 


2239813 


27082081 


25 


2339619 


31597535 


25 


2082495 


24989586 


26 


2192555 


29404980 


26 


1935916 


23053670 


27 


2054513 


27350467 


27 


1799362 


21254308 


28 


1924949 


25425518 


28 


1672168 


19582140 


29 


1803355 


23622163 


29 


1553699 


18028441 


30 


1689133 


21933030 


30 


1443229 


16585212 


31 


1581924 


20351106 


31 


1340233 


15244979 


32 


1481274 


18869832 


32 


1244463 


14000516 


33 


1386727 


17483105 


33 


1155074 


12845442 


34 


1297957 


16185148 


34 


1071790 


117736517 


35 


1214537 


14970611 


35 


9941012 


107795505 


36 


1136192 


13834419 


36 


9216904 


98578601 


37 


1062597 


127718219 


37 


8541876 


90036.725 


38 


9934239 


117783980 


38 


7912224 


82124501 


39 


9283832 


108500148 


39 


7324573 


74799928 


40 


8672666 


99827482 


40 


6776647 


68023281 


41 


8097906 


91729576 


41 


6265268 


61758013 


42 


7557570 


84172006 


42 


5788268 


55969745 


43 


7049035 


77122971 


43 


5342746 


50626999 


44 


6570821 


70552150 


44 


4927130 


45699869 


45 


6120651 


64431499 


45 


4538952 


41160917 


46 


5696794 


58734705 


46 


4176376 


36984541 


47 


5297801 


53436904 


47 


3837886 


33146655 


48 


4922040 


48514864 


48 


3521760 


29624895 


49 


4567885 


43946979 


49 


3226344 


26398551 


50 


4234369 


39712610 


50 


2950636 


23447915 


51 


3919986 


35792624 


51 


2693109 


20754806 


52 


3623608 


32169016 


52 


2452639 


18302167 


53 


3344112 


28824904 


53 


2228115 


16074052 


54 


3080667 


25744237 


54 


2018730 


14055322 



JOINT LIVES— EQUAL AGES. 



123 



TABLE NO XIX. (Concluded) 
American Experience Table, Makeham's Formula, 
Annuity Commutation Columns. 



X 


O^KH 


Q^HH 


X 


Q^EH 


n*jKH 


> 


2 <' ,Q ^ 


8<S i 


> 




^-CD c 2 
<^ ™ ro CD 


CD 


-d-r 


^-r 


CD 


h|j £-<-<D 


^ t- CD 




* >% 


8>3 




% >r. 






<w % 


era % 




arq < 


crq < 




a> m 


cd m 




>- CD CD 


CD CD 




CO 


CO 




ca ax 


CO CO 




Dxx 


Nxx 


55 


Dxxx 


Nxxx 


55 


2832246 


J22911991 


J1823485 


12231837 


56 


2598092 


'20313899 


56 


11641619 


10590218 


57 


2377367 


117936532 


57 


|1472446 


9117772 


58 


2169394 


115767138 


58 


11315217 


7802555 


59 


1973567 


113793571 


59 


11169398 


6633157 


60 


1789279 


|12004292 


60 


|1034424 


55987326 


61 


1616105 


|10388187 


61 


| 9098670 


46888656 


62 


1453492 


| 8934695 


62 


| 7952205 


38936451 


63 


1301208 


l 7633487 


63 


| 6902130 


32034321 


64 


1158826 


6474661 


64 
65 


| 5944088 


26090233 


65 


1026092 


54485694 


| 5074946 


21015287 


66 


9027165 


45458529 


66 


4291171 


16724116 


67 


7885623 


37572906 


67 


| 3590009 


13134107 


68 


6834194 


30738712 


68 


2968023 


10166084 


69 


5870864 


24867848 


69 


2421490 


7744598 


70 


4994331 


19873517 


70 


1946890 


5797704 


71 


4202600 


15670917 


71 


| 1539917 


4257787 


72 


3493904 


12177013 


72 


1196139 


3061648 


73 


2866141 


9310872 


73 


9106590 


2150989 


74 


2316369 


6994503 
5153337 


74 


6779780 


14730113 


75 


1841166 


75 


4923099 


9807014 


76 


1436771 


3716566 


7-6 


3477559 


6329455 


77 


1098558 


26180077 


77 


2382443 


3947012 


78 


8209840 


| 17970237 


78 


1577192 


2369820 


79 


5983045 


; 11987.192 


79 


1005451 


13643691 


80 


4239284 


7747908 


80 


6144843 


7498848 


81 


2911470 


4836438 


81 


3583730 


3915118 


82 


1931399 


2905039 


82 


1984126 


1930992 


83 


1232765 


1672274.3 


83 


1036753 


8942393 


84 


753858.6 


918415.7 


84 


5080253 


3862140 


85 


439733.7 


478682.0 


85 


2319156 


15429841 


86 


243195.9 


235486.1 


86 


9774044 


5655797 


87 


126827.7 


108658.41 


87 


3771846 


1883951 


88 


61958.34 


46700.07 


88 


1319716 


5642354 


89 


28143.49 


18556.58 


89 


4139906 


1502448 


90 


11799.47 


6757.109 


90 


1151628 


3508199 


91 


4520.172 


2236.937 


91 


2797986 


710212.6 


92 


1571.549 


665.3880 


92 


587759.3 


122453.3 


93 


490.0300 


175.3580 


93 


104866.6 


17586.69 


94 


134.7628 


40.5952 
7.9457 


94 


15498.84 


2087.852 


95 


32.6496 


95 


1893.654 


194.198 


96 


6.7384 


1.2073 


96 


181.9361 


12.262 


97 


1.0652 


.1421 


97 


11.7170 


.5446 


98 


.1341 


.0080 


98 


.5366 


.0080 


99 


.0080 


.0000 


99 


.0080 


.0000 



124 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XX. 
American Experience Table, Makeham's Formula, 
Annuity Commutation Columns. 



> 


Two Lives 
Equal Ages 
Six Per 
Cent 


Two Lives 
Equal Ages 
Six Per 
Ceni; 


X 

> 

OR? 

ct> 


Three Lives 
Equal Ages 
Six Per 
Cent 


Three Lives 
Equal Ages 
Six Per 
Cent 




Dxx 


Nxx 




Dxxx 


Nxxx 


10 


5592904 


71118805 


10 


5597526 


164268432 


11 


5195952 


65922853 


11 


5160360 


59108072 


12 


4826910 


61095943 


12 


4757064 


54351008 


13 


4484004 


56611939 


13 


4385177 


49965831 


14 


4165387 


52446552 


14 


4042258 


45923573 


15 


3869265 


48577287 


15 


3725947 


42197626 


16 


3594064 


44983223 


16 


3434200 


38763426 


17 


3338312 


41644911 


17 


3165120 


35598306 


18 


3100645 


38544266 


18 


2916962 


32681344 


19 


2879732 


35664534 


19 


2688029 


29993315 


20 


2674459 


32990075 


20 


2476923 


27516392 


21 


2483673 


30506402 


21 


2282197 


25234195 


22 


2306361 


28200041 


22 


2102593 


23131602 


23 


2141529 


26058512 


23 


1936888 


21194714 


24 


1988363 


24070149 


24 

25 


1784079 


19140635 


25 


1845997 


22224152 


1643122 


17767513 


26 


1713640 


20510512 


26 


1513059 


16254454 


27 


1590601 


718919911 


27 


1393065 


14861389 


28 


1476235 


17443676 


28 


1282376 


13579013 


29 


1369906 


16073770 


29 


1180283 


12398730 


30 


1271062 


14802708 


30 


1086021 


11312709 


31 


1179159 


13623549 


31 


9990775 


103136315 


32 


1093718 


12529831 


32 


9188652 


93947663 


33 


1014248 


115155827 


33 


8448181 


85499482 


34 


9403679 


105752148 


34 


7765088 


77734394 


35 


8716298 


97035850 


35 


7134275 


70600119 


36 


8077100 


88958750 


36 


6552225 


64047894 


37 


7482636 


81476114 


37 


6015066 


58032828 


38 


6929551 


74546563 


38 


5519110 


52513718 


3.9 


6414773 


68131790 


39 


5061000 


47452118 


40 


5935947 


62195843 


40 


4638230 


42814888 


41 


5490267 


56705577 


41 


4247765 


38566723 


42 


5075588 


51629989 


42 


3887333 


34679390 


43 


4689401 


46940588 


43 


3554284 


31125106 


44 


4330028 


42610560 


44 


3246872 


27878234 


45 


3995324" 


38615236 • 


45" 


"2962852" 


24915382 


46 


3683565 


34931671 


46 


2700458 


22214924 


47 


3393259 


31538412 


47 


2458179 


19756745 


48 


3122840 


28415572 


48 


2234424 


17522321 


49 


2870810 


25544762 


49 
50 


2027678 


15494643 


50 


2636090 


22908672 


1836907 


13657736 


51 


2417351 


20491321 


51 


1660783 


11996953 


52 


2218605 


18272716 


52 


1498209 


10498744 


53 


2023498 


| 16249218 


I 53 


1348216 


91505280 


54 


1846504 


14402714 


| 54 


1209995 


79405330 



JOINT LIFE COMMUTATION COLUMN?. 



125 



TABLE NO. XX. (Concluded) 
American Experience Table, Makeham's Formula, 
Annuity Commutation Columns. 



X 


owa^ i 


OcgKH 


X. 


Qw&^ 




nw s H 


> 


g ffg < 


3 * c 3 


> 


3 * C « 




3 * c - 


CD 


CD 










"►3" ' 


">1 




■* >c 




** >£ 




w g 


crq p 




on? <j 




tw <« 




cc M 


<r ^ 




CD 05 




Ct CD , 




03 


co 




CO CO 




CO CO J 




Dxx 


Nxx 




Dxxx 




Xxxx j8 


55 


1681550 1: 


L2721164 


55 


1082657 




68578760 


56 


1528012 


L1193152 


56 


9655051 




58923709 


57 


1383007 


98081450 


57 


8578180 




50345529 


58 


1253365 


85547800 


58 


7589907 




42755622 


59 


1128170 


74266100 


59 
60 


| 6684746 
5857366 




36070876 


60 


1013175 


64134348 


30213510 


61 


9064817 


55069531 


61 


5103492 




25110018 


62 


8075802 


46993729 


62 


4418353 




20691665 


63 


7161485 


39832244 


63 


3798739 




16892926 


64 


6317681 


33514563 


64 


3240592 




13652334 


65 


5541271 


27973292 


65 


2740658 




10911676 


66 


4829009 


23144283 


66 


2295512 




8616164 


67 


4178555 


18965728 


67 


1902329 




6713835 


68 


3587243 


15378485 


68 


1557903 




5155932 


69 


3052525 


12325962 


69 

70 


i 1259043 




3896889 


70 


2572277 


97536850 


1002725 




28941639 


71 


2144085 


76096000 


71 


7856355 




21085284 


72 


1765706 


58438940 


72 


6044894 




15040390 


73 


1434789 


44091050 


73 


4558756 




10481634 


74 


1148634 


33604705 


74 


3361939 




7119695 


75 


9043792 


24560913 


75 


2418219 




4701476 


76 


6990827 


17570086 


76 


1692060 




3009416 


77 


5294774 


12275312 


77 


1148276 




18611345 


78 


3919607 


8355705 


78 


7529957 ■ 




11081438 


79 


2823017 


5532688 


79 


4755016 




6326422 


80 


1985944 


3546744 


80 


2878626 




3447796 


81 


1351048 


2195696.3 


81 


1663005 




17847906 


82 


887796.7 ! 


1307899.6 


82 


9120335 




8727571 


83 


561313.1 


746586.5 


83 


4720643 




4006928 


84 


340015.1 


406571.4 


84 


2291362 




1715566 


85 


196463.0 1 


210108.4 


85 


1036148 




6794182 


86 


107629.7 I 


102478.65 


86 


4325635 




2468547 


87 


55599.59 


46879.06 


87 


1653532 




8150146. 


88 


26905.64 


19973.43 


88 


5730900 




2419246. 


80 


12106.09 1 


7867.344 


89 
90 


1780806 
490705 




638440.3 


90 


5027.724 ! 


2839.620 


.9 


147734.4 


91 


1907.866 | 


931.7543 


91 


118096.8 


29637.61 


92 


657.0566! 


274.6977 


92 


24573.95 


5063.663 


93 


202.9468! 


71.7509 


93 


4343.063 


720.6004 


94 


55.2898' 


16.4611 


94 
95 


635.8321 
76.9531 


84.7683 


95 


13.2678! 


3.1933 


7.8151 


9R 


2.7125' 


.4808 


96 


7.3237 


.4915 


97 


.4247! 


.0561 


97 




.4672 


.0243 


98 


.0530! 


.0031 


98 




.0212 


.0031 


99 


,0031! 


.0000 


99 




,0031 


.0000 



126 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXI 

Actuaries Table, Makeharn's Formula 

Annuity Commutation Columns. 





Olfld 


Q^EIH 


X 


o^iaH 


Q*lSM 




CD O & < 


2 6 & 3 


«-»■ >-t p u 


3 c £ o 

■ c+ -S p ° 


X 


3 e C g 
c* 4 p u 


<*■ '-i p ^ 


> 


> 


~F 


&F 


Orq 


~F 


~F 


cr<5 


S^S" 


^HS 




£>? 


*>$ 




*93 


*«3 




"S3 


*S8 




03 


CD 




CQ 


02 




Dxx 


Nxx 




Dxxx 


Nxxx 


10 


6744128 


113666335 


57 


3914373 


29884788 


11 


6410221 


107256114 


58 


3581148 


26303640 


12 


6078538 


101177576 


59 


3264994 


23038646 


13 


5763708 


95413868 


60 


2965543 


20073103 


14 


5464865 


89949003 


61 


2682371 


17390732 


15 


5181185 


84767818 


62 


2415288 


14975444 


16 


4902848 


79864970 


63 


2164030 


12811414 


17 


4656223 


75208747 


64 


1928400 


10883014 


18 


4413505 


70795242 


65 


1708235 


9174779 


19 ! 


4183050 


66612192 


66 


1503372 


7671407 


20 


3964229 


62647963 


67 


1313675 


6357732 


21 


3756442 


58891521 


68 


1138984 


52187485 


22 


3559116 


55332405 


69 


9791134 


42396351 


23 


3371710 


51960695 


70 


8338383 


34057968 


24 


3193709 


48766986 


71 


7028753 


27029215 


25 


3024628 


45742358 


72 


5858681 


21170534 


26 


2864000 


42878358 


73 


4823737 


16346797 


27 


2711389 


40166969 


74 


3918533 


12428264 


28 


2566375 


37600594 


75 


3136641 


9291623 


29 


2428564 


35172030 | 


76 


2470590 


6821033 


30 


2297583 


32874447 1 


77 


1911900 


4909133 


31 


12173072 


30701375 1 


1 78 


1451216 


3457917 


32 


12054694 


28646681 


I 79 


1078458 


23794586 


33 


11942130 


26704551 1 


80 


7830811 


15963775 


34 


1835074 


24869477 | 


1 81 


5543537 


10420238 


35 


11733240 


23136237 | 


I 82 


3816787 


6603451 


36 


11636355 


21499882 


I 83 


2549145 


4054306 


37 


11544159 


19955723 1 


I 84 


1685088 


2369218 


38 


11456407 


18499316 1 


1 85 


1025674 


13435438 


39 


|1372868 


17126448 | 


86 


6138344 


7297094 


40 I 


11293322 


15833126 I 


I 87 


3516436 


3780658 


41 


11217535 


14615591 I 


1 88 


1920266 


1860391.6 


42 


11145392 


13470199 1 


1 89 


995072.7 


865318.9 


43 


11076625 


12393574 I 


1 90 


486880.5 


378438.4 


44 


|1011094 


1138248021 


1 91 


223716.7 


154721.72 


45 | 


! 9486278 


1043385241 


92 


95960.80 


58760.92 


46 


I 8890758 


95447766 


1 93 


38174.43 


20586.49 


47 


I 8322879 


87124887 


1 94 


13983.97 


6602.517 


48 


1 7781446 


79343441 


1 95 


4680.243 


1922.274 


49 


| 7265047 


72078394 


| 96 


1418.942 


503.3322 


50 


I 6772563 


653058311 


1 97 


386.0507 


117.28147 


51 


1 6302913 


59002918 


1 98 


93.29245 


23.98902 


52 


1 5855097 


53147821 


1 99 


19.70997 


4.27905 


53 


5428205 


47719616 


1100 


3.629225 


.64982 


54 


5021400 


42698216 


|101 


.565410 


.08441 


55 


4633932 


38064284 


1102 


.076024 


.00839 


56 


4265123 


33799161 


[103 


.008386 


.00000 



JOINT LIFE COMMUTATION COLUMNS. 



127 



TABLE NO. XXII. 

Actuaries Table. Makeham's Formula 

Annuity Commutation Columns. 



x 

> 

CO 


Three 
Equal 
Four 
Cent 


Three 
Equal 
Four 
Cent 


X 

> 

ON? 

1 CO 


Three 
Equal 
Four 
Cent 


Three 
Equal 
Four 
Cent 




Lives 
Ages 
Per 


Lives 
Ages 
Per 


Lives 
Ages 
Per 


Lives 
Ages 
Per 




Dxxx 


Nxxx 




Dxxx 


Nxxx 


10 


6761692 


100738977 


57 


2368336 


14469822 


11 


6367851 


94371126 


58 


2113490 


12356332 


12 


5996512 


88374614 


59 


1876321 


10480011 


13 


5646375 


82728239 


60 


1656342 


8823669 


14 


5316215 


77412024 


61 


1453099 


7370570 


15 


5004880 


72407144 


62 


1266155 


6104415 


16 


4711284 


67695860 


63 


1095078 


50093373 


17 


4434398 


63261462 


64 


9394259 


40099114 


18 


4173261 


59088201 


65 


7987361 


32711753 


19 


3926959 


55161242 


66 


6725117 


25986636 


20 


3694636 


51466506 


67 


5602074 


20384562 


21 


3475482 


47991024 


68 


4612218 


15772344 


22 


3268737 


44722287 


69 


3748864 


12023480 


23 


3073680 


41648607 


70 


3004620 


9018860 


24 


2889636 


38758971 


71 


2371325 


6647535 


25 


2715968 


36043003 


72 


1840348 


4807187 


26 


2552072 


33490931 


73 


1402143 


3405044 


27 


2397383 


31093548 


74 


1046933 


23581110 


28 


2251370 


28842178 


75 


7646230 


15934880 


29 


2113532 


26728646 


76 


5450888 


10483992 


30 


1983384 


24745262 


76 


3784264 


6699728 


31 


1860492 


22884770 


78 


2552103 


4147625 


32 


1744432 


21140338 


79 


1667330 


2480295 


33 


1634811 


19505527 


80 


1052068 


14282268 


34 


1531254 


17974273 


81 


6390440 


7891828 


35 


1433413 


16540860 


82 


3723179 


4168649 


36 


1340961 


15199899 


83 


2072408 


2096241 


37 


1253587 


13946312 


84 


1097322 


9989193 


38 


1171002 


12775310 


85 


5500861 


4488332 


39 


1092932 


11682378 


86 


2597230 


1891102 


40 


1019124 


106632544; 


87 


1148429 


7426727 


41 


9493376 


97139168 


88 


4726168 


2700559 


42 


8833476 


883056921 


89 


1797901 


9026576 


43 


8211357 


80094335! 


90 


6275288 


2751288 


44 


7619398 


72474937J 


91 


1993263 


7580247 


45 


7061451 


65413486! 


92 


5710505 


1869742 


46 


6533938 


588795481 


93 


1461198 


4085437 


47 


6035286 


52844262 


24 


3303792 


781644.7 


48 


5564045 


472802171 


95 


652358.3 


129286.4 


49 


5118874 


42161343 


96 


111057.3 


18229.09 


50 


I 4698546 


37462797 ; 


97 


16072.51 


2156.579 


51 


4301918 


33160879! 


98 


1949.168 


209.4118 


52 


3927968 


29232911' 


99 


192.8319 


16.57994 


53 


3575753 


256571581 


100 


15.53774 


1.04220 


54 


3244422 


22412736! 


|101 


.99137 


.05082 


55 


2933198 


194795381 


102 


.04899 


.00183 


56 


2641380 


168381581 


103 


.00183 


.00000 



128 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXIII. 
Actuaries Table, Makeham's Formula 
Annuity Commutation Columns. 



X 


Q^HH 


O^dHH 


X 


O^HH 


O^&H 


> 


2.® g ° 


5.® g o 


> 


2-® g o 


2® g o 


CD 


-d-r 


►tj&tH 


CD 


►u^r 


>U~ f 










%>% 


%fc< 




crq o 


crq a) 




crq (D 


crq cd 




<x> co 


cd en 




CD CO 


CD CO 




CO 


CO 




CO 


CO 




Dxx 


Nxx 




Dxx 


Nxx 


10 


6142783 


89619008 


57 


2268675 


18521737 


11 


5769756 


83849252 


58 


2055780 


14197282 


12 


5419105 


78430147 


59 


1856439 


12340843 


13 


5089493 


73340654 


60 


1670095 


10670748 


14 


4779649 


68561005 


61 


1496254 


9174494 


15 


4488380 


64072625 


62 


1334441 


7840053 


16 


4214567 


59858058 


63 


1184260 


6655793 


17 


3957155 


55900903 


64 


1045240 


56105527 


18 


3715150 


52185753 


65 


9170840 


46934687 


19 


3487676 


48698077 


1.66 


7994163 


38940524 


20 


3273705 


45424372 


67 


6918927 


32021597 


21 


3072569 


42351803 


68 


5941722 


26079875 


22 


2883440 


39468363 


69 


5059082 


21020793 


23 


2705534 


36762829 


70 


4267412 


16753381 


24 


2538355 


34224474 


71 


3562912 


13190469 


25 


2381070 


31843404 


72 


2941512 


10248957 


26 


2233151 


29610253 


73 


2398824 


7850133 


27 


2094020 


27516233 


74 


1930112 


5920021 


28 


1963149 


25553084 


75 


1530270 


4389751 


29 


1840039 


23713045 


76 


1193842 


31959092 


30 


1724219 


21988826 


77 


9150744 


• 22808348 


31 


1615249 


20373577 


78 


6879662 


15928686 


32 


1512713 


18860864 


79 


5063867 


10864819 


33 


1416227 


17444637 


80 


3641917 


7222902 


34 


1325412 


16119225 


81 


2553608 


4669294 


35 


1239938 


14879287 


82 


1741479 


2927815 


36 


1159479 


13719808 


83 


1151992 


1775823 


37 


1083737 


12636077 


84 


737092.0 


1038731.4 


38 


1012410 


116236666 


85 


454728.9 


584002.5 


39 


9452491 


106784175 


86 


269549.6 


314452.9 


40 


8819946 


97964229 


87 


152944.6 


161508.26 


41 


8224265 


89739964 


88 


82725.04 


78783.22 


42 


7663091 


82076873 


89 


42459.45 


36323.77 


43 


7134426 


74942447 


90 


20577.18 


15746.586 


44 


6636348 


68306099 


91 


9364.968 


6381.618 


45 


6167055 


62139044 


92 


3978.743 


2402.875 


46 


5724858 


56414186 


93 


1567.719 


835.1560 


47 


5308185 


51106001 


1 94 


568.8138 


266.3422 


48 


4915575 


46190426 


I 95 


188.5611 


77.78111 


49 


4545654 


41644772 


| 96 
| 97 


57.94189 


19.83922 


50 


4197156 


37447616 


15.25867 


4.58055 


51 


3868899 


33578717 


I 98 


3.652267 


.92828 


52 


3559786 


30018931 


| 99 


.7642677 


.16401 


53 


3268808 


26750123 


|100 


.1393856 


.02462 


54 


2995043 


23755080 


|101 


.0217582 


.00286 


55 


2737612 


21017468 


1102 


.0028645 


.00000 


56 


2495731 


18521737 


|103 


1 


1 



JOINT LIFE COMMUTATION COLUMNS. 



129 



TABLE NO. XXIV. 

Actuaries Table, Makeham's Formula 

Annuity Commutation Columns. 



X 


C^BH O^.^H 


X 


O^KH 


Q^HH 


> 




2 ~'»2 V 


> 


**" w M (D 




a> 


hj~o 


l-tf£l o 


CO 


►"tfS-CD 


HJ&CD 




82 












Dxxx 


Xxxx 




Dxxx 


Nxxx 


10 


6144631 


80501337 


57 


1372630 


7948018 


11 


5731620 


74769717 


58 


1213262 


6734756 


12 


5345974 


69423743 


59 


1066851 


56679048 


13 


4985884 


64437859 


60 


9328081 


47350967 


14 


4649638 59788221 


61 


8105533 


39245434 


15 


4335651 155452570 


62 


6995479 


32249955 


16 


4042443 151410127 


63 


5992660 


26257295 


17 


3768630 (47641497 


64 


5091914 


21165381 


18 


3512920 144128577 


65 


4288110 


16877271 


19 


3274109 140854468 


66 


3576073 


13301198 


20 


3051072 |37803396 


62 


2950527 


10350671 


21 


12842758 134960638 


68 


2406050 


7944621 


22 


2648188 


32312450 


69 


1937040 


6007581 


23 


2466446 


29846004 


70 


1537703 


4469878 


24 


2296683 


27549321 


71 


1202064 


32678137 


25 1 


2138088 125411233 


72 


9239972 


23458165 


26 i 


1989930 


23421303 


73 


6972801 


16465364 


27 | 


1851513 


21569790 


74 


5156770 


11308594 


28 | 


1722186 


19847604 


75 


3730357 


7578237 


29 1 


1601346 


18246258 


76 


2633990 


4944247 


30 1 


1488429 


16757829 || 77 


1811222 


3133025 


31 1 


1382908 


15374921 I! 78 


1209855 


19231699 


32 | 


1284292 


14090629 |! 79 


7828902 


11402797 


33 1 


1192121 


12898508 1! 80 


4892907 


6509890 


34 H1105972 


11792536 |! 81 


2943730 


3566160 


35 


11025443 110767092811 82 


1698733 


18674272 


36 1 


1 9501707! 9816922111 83 


9365491 


9308781 


37 1 


I 8798000! 89371221'! 84 


4911718 


4397063 


38 


i 81401241 8123109711 85 


2438738 


1958325 


39 | 


| 7525074! 73706023H 86 


1140507 


8178184 


40 | 


I 6950060' 66755963 ' 87 


4995003 


3183181 


41 ! 


1 6412484' 60343479M 88 


2036032 


11471490 


42 


I 5909916 1 5443356311 89 


7671589 


3799901 


43 


| 5440121! 48993442!! 90 


2652145 


11477561 


44 1 


| 5001017! 43992425|l 91 


8343963 


3133598 


45 


! 4590669! 39401756!! 92 


2367699 


765899.5 


46 


1 4206790! 351949661! 93 


6000742 


165825.3 


47 


! 3849175' 31345791!! 94 


1343855 


31439.79 


48' 


I 3514832! 278309591! 95 


262827.0 


5157.086 


49 


! 3202819! 24628140|| 96 


44317.50 


725.3360 


50 


| 2911824' 21716316" 97 


6352.662 


90.06981 


51 


! 2640634' 19075682" 98 


762.2886 


13.84095 


52 


1 23881311 16687551!' 99 


74.77195 


6.36375 


53 


I 2153288! 14534263)1100 


5.96749 


.39626 


54 


! 1935154! 12599109'!101 


.37713 


.01913 


55 


I 1732861! 10866248H102 


.01846 


.00067 


56 


! 1545600 


9320648 


1103 


.00067 


.00000 



130 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXV. 
Actuaries Table, Makeham's Formula 
Annuity Commutation Columns. 



9< 

> 
OP 


Two Lives 
Equal Ages 
Six Per 

Cent 


Two Lives 
Equal Ages 
Six Per 
Cent 


X 

> 

orq 


Two Lives 
Equal Ages 
Six Per 
Cent 


Two Lives 
Equal Ages 
Six Per 
Cent 




Dxx 


Nxx 




Dxx 


Nxx 


10 


5587153 


71739261 


57 


1321685 


8910336 


11 


5198467 


66540794 


58 


1186358 


7723978 


12 


4836449 


61704345 


69 


1061216 


66627624 


13 


4499447 


57204898 


60 


9456896 


57170728 


14 


4185661 


53019237 


61 


8392567 


48778161 


15 


3893510 


49125727 


62 


7414336 


41363825 


16 


3623164 


45502563 


63 


6517696 


34846129 


17 


3368227 


42134336 


64 


5698431 


29147693 


18 


3132404 


39001932 


65 


4952590 


24195108 


19 


2912832 


36089100 


66 


4276413 


19918695 


20 


2708372 


33380728 


67 


3666308 


16252387 


21 


2517988 


30862740 


68 


3118787 


13133600 


22 


2340704 


28522036 


69 


2630442 


10503158 


23 


2175615 


26346421 


70 


2197885 


8305273 


24 


2021877 


24324544 


71 


1817728 


6487545 


25 


1878706 


22445838 


72 


1486545 


5001000 


26 


1745369 


20700469 


73 


1200851 


38001491 


27 


1621188 


19079281 


74 


9570988 


28430503 


28 


1505530 


17573751 


75 


7516667 


20913836 


29 


1397805 


16175946 


76 


5808827 


15105009 


30 


1297464 


14878482 


77 


4410418 


10694591 


31 


1203998 


13674484 


78 


3284542 


7410049 


32 


1116931 


12557553 


79 


2394823 


5015226 


33 


1035821 


115217320 


80 


1706099 


3309127 


34 


9602584 


105614736 


81 


1184981 


2124145.7 


35 


8898571 


96716165 


82 


800480.4 


1323665.3 


36 


8242643 


88473522 


83 


524535.4 


799129.9 


37 


7631474 


80842048 


84 


340196.8 


458933.1 


38 


7061983 


73780065 


85 


203162.9 


255770.2 


39 


6531310 


67248755 


86 


119292.8 


136477.35 


40 


6036786 


61211969 


87 


67049.08 


69428.27 


41 


5575937 


55636032 


88 


35923.56 


33504.71 


42 


5146456 


50489576 


89 


18264.19 


15240.522 


43 


4746209 


45743367 


90 


8767.882 


6472.640 


44 


4373210 


41370157 


91 


3952.747 


2519.893 


45 


4025611 


37344546 


92 


1663.495 


856.3983 


46 


3697453 


33647093 


93 


515.7365 


340.6618 


47 


3399911 


30247182 


94 


233.3530 


107.30881 


48 


3118739 


27128443 


95 


76.62632 


30.68249 


49 


2856832 


24271611 


96 


22.79303 


7.88946 


50 


2612934 


21658677 


97 


6.08428 


1.80518 


51 


2385847 


19272830 


98 


1.44258 


.36261 


52 


2174518 


17098312 


99 


.29902 


.06358 


53 


1977937 


15120375 


100 


.05402 


.00956 


54 


1795141 


13325234 


101 


.00835 


.00121 


55 


i 1625402 


11699832 


102 


.00109 


.00012 


66 


1 1467811 


10232021 


103 


.00012 


.00000 



JOINT LIFE COMMUTATION COLUMNS. 



131 



TABLE NO. XXVI. 

Actuaries Table, Makeham's Formula 
Annuity Commutation Columns. 



X 


OMHH 


CMRH 


X 


O^HH 


OCfi HH 


> 


3 * C "l 


CD 5* -2 ~ 
3 *'C ^ 


> 


3 * c -i 


3 ^ £ <-t 


CD 


~V?L% 


~^£.g 


era 

CD 


^gig 


*>*&% 




8 >r 


5 >r 




* >t- 


* >F 






8| 




*2 


era 3- 

*3 




Dxxx 


Nxxx 




Dxxx 


Nxxx 


10 


5588949 


65164416 


56 


9090115 


51971763 


11 


5164107 


60000309 


57 


7996670 


43975093 


12 


4771209 


55229100 


58 


7001543 


36973550 


13 


4407851 


50821249 


59 


6098572 


30874978 


14 


4071872 


46749377 


60 


5282004 


25592974 


15 


3761022 


42988355 


~61 


4546436 


21046538 


16 


3473593 


39514762 


62 


3886783 


17159755 


17 


3207760 


36307002 


63 


3298192 


13861563 


18 


2961898 


33345104 


| 64 


2776008 


11085555 


19 


2734503 


30610601 


i 65 


2315737 


8769818 


20 


2524184 


28086417 


! 66 


1912989 


6856829 


21 


2329657 


'25756760 


67 


1563471 


5293358 


22 


12149732 


23607028 


I 68 


1262926 


4030432 


23 


I198331Q 121623718 


69 


1007155 


30232765 


24 


1829374 |19794344 


| 70 


7919778 


22312987 


25 1 


1686986 118107358 


71 


6132699 


16180288 


26 i 


1555274 116552084 


1 72 


4669583 


11510705 


27 1 


H433440 115118644 


73 


3490582 


8020123 


28 | 


11320736 113797908 


74 


2557128 


5462995 


29 | 


1216480 112581428 


75 


1832348 


3630647 


30 1 


11120034 111461394 


76 


1281609 


23490377 


31 1 


'1030812 1104305822 


77 


8729656 


14760721 


32 1 


1 9482728' 94823094 


78 


5776185 


8984536 


33 1 


I 87191491 86103945 


79 


3702474 


5282062 


34 | 


I 8012730! 78091215 


80 
81 


2292141 


2989921 


35 ! 


73592421 70731973 


1366014 


16239069 


36 1 


! 6754688! 63977285 


82 


7808482 


8430587 


37 1 


1 6195424! 57781861 


83 


4264377 


4166210 


38 1 


1 56780801 52103781 


84 


2215347 


1950863 


39 1 


! 5199539! 46904242 


85 


1089597 


8612655 


40 | 


4756922! 42147320 


86 


5047467 


3565188 


41 


1 43475751 37799745 


87 


2189749 


13754388 


42 ! 


3965040 1 33S34705 


88 


8841522 


4912866 


43 1 


1 36190641 30215641 


Eii 


3299981 


1612885 


44 | 


I 3295562! 26920079 


90 


1130072 


4828132 


45 


1 29966101 23923469 


91 


3521804 


1306328.4 


46 1 


1 27173081 21206161 


92 


989924.5 


316403.9 


47 ! 


1 2465410! 18740751 


93 


248521.8 


67882.11 


48 1 


! 22300231 16510728 


94 


55130.96 


12751.15 


49 |! 2012893! 14497835 


95 


10680.60 


2070.550 


50 II 1812475 


12685090 


9C 


1783.961 


286.5886 


51 l| 1628408 


1 11056682 


97 


253.3077 


33.28092 


52 1 1458803 


9597879 


98 


30.10894 


3.17198 


53 I! 1302939 


8294940 


99 


2.92549 


.24649 


54 II 1159901 


7135039 


100 


.23128 


.01521 


55 || 1028851 


6106187S 


101 


.01448 


.00074 








102 


.00074 


.00000 



132 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXVII. 
Showing the Force of Mortality, ^, at each year of age ac- 
cording to the Makehamized, Carlisle, Actuaries and Amer- 
ican Experience Tables of Mortality. 



> 

OK} 


> 
B 

"-i 

o' 
p 
p 


O 

p 

w 

CO 


Actuaries 


> 
ere 


> 

B 

(V 

<s 

p 



a 

p 

w 


> 
a 

p 


10 
11 
12 
13 
14 
15 
16 
17 
18 
19 


.00768 
.00769 
.00770 
.00772 
.00773 
.00775 
.00776 
.00778 
.00781 
.00783 


.00495 
.00507 
.00519 
.00533 
.00548 
.00565 
.00583 
.00602 
.00624 
.00648 


.00693 
.00695 
.00698 
.00700 
.00703 
.00707 
.00710 
.00714 
.00719 
.00724 


59 

60 

61 

62 

63 « 

64 

65 

67 
68 


.02373 
.02553 
.02752 
.02974 
.03220 
.03494 
.03798 
.04136 
.04512 
.04929 


.02632 
.02804 
.02993 
.03199 
.03425 
.03672 
.03942 
.04238 
.04562 
.04917 


.02671 
.02869 
.03086 
.03323 
.03582 
.03867 
.04179 
.04519 
.04892 
.05301 


20 

21 
22 
23 
24 
25 
26 
27 
28 
29 


.00786 
.00788 
.00792 
.00793 
.00799 
.00804 
.00809 
.00814 
.00821 
.00827 


.00674 
.00702 
.00733 
.00767 
.00804 
.00894 
.00902 
.00910 
.00920 
.00930 


.00729 
.00735 
.00741 
.00748 
.00755 
.00764 
.00773 
.00783 
.00793 
.00805 


69 

70 
71 

72 
73 

74 

75 
67 
77 
78 


.05393 
.05908 
.06481 
.07117 
.07824 
.08610 
.09483 
.10453 
.11531 
.12729 


.05305 
.05730 
.06195 
.06703 
.07260 
.07870 
.08537 
.09267 
.10066 
.10941 


.05748 
.06239 
.06775 
.07362 
.08005 
.08708 
.09479 
.10322 
.11046 
.12057 


30 
31 
32 
33 
34 
35 
36 
37 
38 
39 


.00835 
.00843 
.00853 
.00863 
.00875 
.00888 
.00902 
.00918 
.00935 
.00955 


.00942 
.00955 
.00968 
.00983 
.01000 
.01018 
.01038 
.01060 
.01083 
.01109 


.00818 
.00833 
.00848 
.00865 
.00884 
.00905 
.00927 
.00952 
.00979 
.01008 


79 

80 

81 
82 
83 
84 
85 
86 
87 
88 


.14060 
.15540 
.17183 
.19010 
.21040 
.23295 
.25801 
.28586 
.31681 
.35120 


.11898 
.12945 
.14092 
.15347 
.16720 
.18223 
.19869 
.21669 
.23640 
.25797 


.13165 
.14374 
.15702 
.16155 
.17746 
.19488 
.21396 
.23485 
.25771 
.28275 


40 
41 
42 
43 
44 
45 
46 
47 
48 
49 


.00977 
.01001 
.01028 
.01058 
.01091 
.01128 
.01169 
.01215 
.01265 
.01321 


.01137 
.01168 
.01202 
.01239 
.01280 
.01324 
.01373 
.01426 
.01484 
.01548 


.01030 
.01069 
.01108 
.01154 
.01204 
.01260 
.01321 
.01382 
.01449 
.01521 


89 
90 
91 
92 
93. 
94 
95 
96 
97 
98 


.38941 
.43187 
.47905 
.53149 
.58975 
.65449 
.72643 
.80637 
.89521 
.99392 


.28158 
.30742 
.33569 
.36665 
.40053 
.43760 
.47818 
.52258 
.57120 
.62440 


.31016 
.34018 
.37304 
.40902 
.44832 
.49146 
.53868 
.59029 
.64701 
.70900 


50 
51 
52 
53 
54 
55 
56 
57 
58 


.01384 
.01458 
.01531 
.01617 
.01712 
.01818 
.01936 
.02066 
.02212 


.01618 
.01690 
.01778 
.01870 
.01970 
.02079 
.02190 
.02331 
.02475 


.01609 
.01697 
.01792 
.01897 
.02012 
.02137 
.02275 
.02426 
.02591 


99 
100 
101 
102 
103 
104 
105 




.68262 
.74635 
.81611 
.89243 
.97599 
1.06742 
1.16751 


.77687 
.78018 
.86154 
.94062 



134 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXVIII. 

Annuities on Two and Three Lives, Equal Ages, American^Ex- 

perience Table — Makehamized 



X 


S'a 


^H 


W H 


^H 


°->M 


05 H 


> 

era 
o 






P 


p 


P V 

i— • 


p 




> 


> 


> 


> 


> 


> 




CO 

02 


era 

CO 


CTQ 

CD 

to 


0*3 
CD 
CO 


era 

CD 
CO 


as 

CD 
CO 




axx 2 Lives 


axxx 3 Lives 


axx 2 Lives 


axx 3 Lives 


axx 2 Lives 


axxx 3 Lives 


10 


18.03023 


15.74490 


14.45673 


19.90393 


12.71569 


11.48156 


11 


17.95018 


15.67560 


14.41467 


12.86509 


12.68735 


11.45624 


12 


17.86687 


15.60359 


14.37032 


12.82409 


12.65736 


11.42527 


13 


17.77960 


15.52764 


14.32333 


12.80909 


12.62531 


11.39410 


14 


17.68582 


15.44761 


14.27348 


12.73382 


12.59104 


11.36087 


15 


17.59277 


15.36397 


14.22090 


12.68452 


12.55379 


11.32534 


16 


17.53350 


15.27584 


14.16537 


12.63228 


12.51597 


11.28774 


17 


17.38916 


15.18358 


14.10654 


12.57680 


12.47185 


11.24716 


18 


17.28050 


15.08675 


14.04486 


12.51811 


12.43105 


11.20389 


19 


17.16730 


14.98554 


13.97948 


12.45610 


12.38467 


11.15811 


20 


17.04899 


14.87918 


13.91044 


12.39014 


12.33523 


11.10910 


21 


16.72565 


14.76780 


13.83767 


12.32054 


12.28278 


11.05697 


22 


16.79700 


14.65129 


13.76100 


12.24678 


12.22707 


11.00144 


23 


16.66317 


14.53996 


13.68026 


12.16909 


12.16828 


10.94266 


24 


16.52351 


14.40209 


13.59509 


12.08721 


12.10568 


10.87992 


25 


16.37807 


14.26878 


13.50542 


11.99908 


12.03911 


10.81326 


26 


16.22758 


14.12986 


13.41129 


11.90740 


11.96839 


10.74278 


27 


16.06985 


13.98500 


13.31226 


11.81213 


11.89484 


10.66812 


28 


15.90695 


13.83382 


13.20857 


11.71065 


11.81633 


10.58895 


29 


15.73641 


13.67593 


13.09901 


11.60356 


11.76026 


10.50487 


30 


15.56053 


13.51244 


12.98479 


11.49146 


11.64593 


10.41666 


31 


15.37792 


13.34189 


12.86478 


11.37399 


11.55628 


10.32315 


32 


15.18800 


13.16442 


12.73872 


11.25999 


11.45618 


10.22430 


33 


14.99176 


12.98058 


12.60746 


11.12073 


11.353*81 


10.12045 


34 


14.78823 


12.78940 


12.46969 


10.98504 


11.24582 


10.01075 


25 


14.57817 


12.59190 


12.32615 


10.84362 


11.13269 


9.89816 


36 


14.36078 


12.38714 


12.17611 


10.69545 


11.01370 


9.77498 


37 


14.13611 


12.17510 


12.01947 


10.54064 


10.88869 


9.64791 


38 


13.90438 


11.95622 


11.85336 


10.37945 


10.75280 


9.51490 


39 


13,69744 


11.73196 


11.68700 


10.21219 


10.62108 


9.37617 


40 


13.42000 


11.49895 


11.51059 


10.03790 


10.47782 


9.23077 


41 


13.16720 


11.25983 


11.32757 


9.85722 


10.32838 


9.07928 


42 


12.90632 


11.01364 


11.13744 


9.66952 


10.17222 


8.92111 


43 


12.64050 


10.76150 


10.94092 


9.47584 


10.00993 


8.75707 


44 


12.35815 


10.50264 


10.73719 


9.27517 


9.84071 


8.58618 


45 


12.08683 


10.23897 


10.52690 


9.06848 


9.66511 


8.40927 


46 


11.80040 


9.96784 


10.31013 


8.85565 


9.50498 


8.22635 


67 


11.50784 


9.69201 


10.08662 


8.63670 


9.29442 


8.03674 


48 


11.20943 


9.41109 


9.85666 


8.41155 


9.09927 


7.84199 


49 


10.90596 


9.12608 


9.62086 


8.18219 


8.89833 


7.64159 


50 


10.59691 


8.83627 


9.37864 


7.94572 


'8.69039 


7.43518 


51 


10.28323 


8.54292 


9.13080 


7.70662 


8.47677 


7.22374 


52 


9.96531 


8.24651 


8.87762 


7.46224 


8.25512 


7.00753 


53 


9.64402 


7.94782 


8.61960 


7.21419 


8.03024 


6.78712 


54 


9.31918 


7.66449 


8.35671 


6.96245 


7.79999 


6.56248 



JOINT LIFE ANNUITIES. 



135 



TABLE NO. XXVIII. 

Annuities on Two and Three Lives, American Experience 
Table, Makehamized. 



X 

> 

ore 

CD 












05 H 




> 

CD 

92 


> 

. ere 

CD 
02 


> 

CD 

92 


> 

ere 

CD 

02 


> 
ere 

CD 
92 


ere 

CD 
0B 




axx 


axxx 


axx 


axxx 


axx 


axxx 




2 Lives 


3 Lives 


2 Lives 


3 Lives 


2 Lives 


3 Lives 


55 


8.99175 


7.34469 


8.08967 


6.70794 


7.56497 


6.33430 


56 


8.66213 


7.04164 


7.81877 


6.43609 


7.32530 


6.10289 


57 


8.33113 


6.73869 


7.54470 


6.19084 


7.08166 


5.86902 


58 


7.99938 


6.43502 


7.26706 


5.93252 


6.82546 


5.63321 


59 


7.66750 


6.13569 


6.98915 


5.67242 


6.58288 


5.39599 


60 


7.33641 


5.83724 


6.70901 


5.41245 


6.34463 


5.15821 


61 


7.00651 


5.54149 


6.42791 


5.15335 


6.07508 


4.92016 


62 


6.67908 


5.24983 


0.14705 


4.90759 


5.81908 


4.68313 


63 


6.35417 


4.96212 


5.86646 


4.64122 


5.56201 


4.44698 


64 


6.03297 


4.67958 


5.58726 


4.38928 


5.30408 


4.21291 


65 


5.71605 


4.40277 


5.31002 


4.14099 


5.04817 


3.98141 


66 


5.40445 


4.13249 


5.03575 


3.89735 


4.79276 


3.75347 


67 


5.09842 


3.86902 


4.76487 


3.65852 


4.53873 


3.52927 


68 


4.79878 


3.61297 


4.49778 


3.42546 


4.28699 


3.30954 


69 


4.50639 


3.36514 


4.23581 


"3.19091 


4.03796 


3.09512 


70 


4.22161 


3.12569 


3.97922 


2.97793 


3.79185 


2.88629 


71 


3.94526 


2.89530 


3.72888 


2.76558 


3.54911 


2.68385 


72 


3.67771 


2.67413 


3.48522 


2.55961 


3.30916 


2.48811 


73 


3.41919 


2.46233 


3.24857 


2.36201 


3.07300 


2.29923 


74 


3.17392 


2.26015 


3.01960 


2.17265 


2.92562 


2.11774 


75 


2.93153 


2.06808 


2.79895 


1.89204 


2.71578 


.1.94419 


76 


2.70297 


1.88589 


2.58675 


1.82009 


2.51331 


1.77851 


77 


2.48464 


7.71345 


2.38312 


1.65671 


2.31841 


1.62081 


78 


2.27719 


1.55129 


2.18888 


1.50256 


2.13177 


1.47165 


79 


2.08007 


1.39930 


2.00353 


1.35697 


1.95534 


1.33049 


80 


1.89376 


1.25009 


1.82765 


1.22035 


1.78592 


1.19772 


81 


1.71804 


1.12260 


1.66117 


1.09247 


1.62514 


1.07323 


82 


1.55285 


.99867 


1.50411 


.97300 


1.47319 


.95693 


83 


1.39813 


.88305 


1.35652 


.86254 


1.33105 


.84881 


84 


1.25364 


.77816 


1.21827 


.76023 


1.19574 


.74861 


85 


1.11851 


.68026 


1.08855 


.66532 


1.06945 


.65572 


86 


.99354 


.59104 


.96827 


.57865 


.95214 


.57068 


87 


.87794 


.51086 


.85682 


.49948 


.84275 


.49278 


88 


.77145 


.43591 


. .75373 


.42695 


.74235 


.42214 


89 


.67411 


.36973 


.65936 


.36292 


.64987 


.35851 


90 


.58219 


.31014 


.57216 


.30533 


.56349 


.30107 


91 


.50496 


.25826 


.49488 


.25383 


.48838 


.25096 


92 


.43165 


.21186 


.42340 


.20834 


.41807 


.20606 


93 


.36456 


.17050 


.35785 


.16771 


.35355 


.16592 


94 


.30659 


.13715 | 


.30121 


.13472 


.29772 


.13332 


95 


.24750 


.10394 J 


.24337 


.10255 


.24068 


.10156 


96 


.18210 


.06883 


.17917 


.06740 


.17727 


.06711 


97 


.13547 


.05359 


.13343 


.04658 


! .13210 


.05203 



136 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXIX. 
Annuities on Two and Three Lives, Actuaries Table Makehamized 



X 


^H 


^H 


cn H 


oi H 


^H 


b»H 


> 






^g 




^S 




CD 


* £ 


p 


P 


P 


P 


P 




> 


> 


> 


> 


> 


> 




CK? 


<w 


CK? 


crcj 


OKJ 


OP 




CD 


cd 


CD 


CD 


CD 


CD 




co 


CO 


CO 


CO 


CO 


CO 




axx 


axxx 


axx 


axxx 


axx 


axxx 




2 Lives 


3 Lives 


2 Lives 


3 Lives 


2' Lives 


3 Lives 


10 


16.84028 


14.89848 


14.59928 


13.10109 


12.84003 


11.65951 


11 


16.73230 


14.81995 


14.53255 


13.04513 


12.80008 


11.61894 


12 


16.64482 


14.73767 


14.47189 


12.98616 


12.75813 


11.57594 


13 


16.55425 


14.65156 


14.41021 


12.92406 


12.71376 


11.52945 


14 


16.45951 


14.56153 


14.34437 


12.85868 


12.66704 


11.48124 


15 


16.36074 


14.46731 


14.27555 


12.78988 


12.61837 


11.42996 


16 


16.28946 


14.36888 


14.20266 


12.71772 


12.58774 


11.37550 


17 


16.15229 


14.26608 


14.12654 


12.64160 


12.50935 


11.32391 


18 


16.04055 


14.15876 


14.04674 


12.56179 


12.47982 


11.28387 


19 


15.92431 


14.04648 


13.96310 


12.47804 


12.38970 


11.19421 


20 


15.80331 


13.93006 


13.87552 


12.39020 


12.32502 


11.12693 


21 


15.67741 


13.80848 


13.78384 


12.29807 


12.25691 


11.05603 


22 


15.54671 


13.68182 


13.68797 


12.20172 


12.18532 


10.98138 


23 


15.41078 


13.51891 


13.58770 


12.10081 


12.10971 


10.90284 


24 


15.26970 


13.41310 


13.48293 


li.99529 


12.03067 


10.82088 


25 


15.12330 


13.27078 


13.37355 


11.88503 


11.94750 


10.73356 


26 


14.97129 


13.12304 


13.25941 


11.76991 


11.86022 


10.64254 


27 


14.81418 


12.96976 


13.14041 


11.64442 


11.76868 


10.54711 


28 


14.65125 


12.81084 


13.01638 


11.52466 


11.67280 


10.44711 


29 


14.48231 


12.64646 


12.88726 


11.39433 


11.57239 


10.34252 


30 


14.30828 


12.47628 


12.75292 


11.25873 


11.46735 


10.23309 


31 


14.12810 


12.30038 


12.61328 


11.11782 


11.35756 


10.11908 


32 


13.94207 


12.11875 


12.46823 


10.97152 


11.24291 


9.99956 


33 


13.75014 


11.93137 


12.31772 


10.81979 


11.12288 


9.87529 


34 


13.55230 


11.73818 


12.16167 


10.66259 


10.99832 


9.74587 


35 


13.34855 


11.53945 


12.00002 


10.50020 


10.86873 


9.61131 


36 


13.13882 


11.33508 


11.83274 


10.33175 


10.73364 


9.47337 


37 


12.92336 


11.12512 


11.65979 


10.15812 


10.59324 


9.32654 


38 


12.70203 


10.90972 


11.48119 


9.95613 


10.44750 


9.17631 


39 


12.47494 


10.68902 


11.29691 


9.79472 


10.29640 


9.02083 


40 


12.24221 


10.46312 


11.10706 


9.60508 


10.13981 


8.86021 


41 


12.00425 


10.23231 


10.91165 


9.41031 


9.97788 


8.69444 


42 


11.76034 


9.99671 


10.71061 


9.21055 


9.81281 


8.52466 


43 


11.51150 


9.75409 


10.50434 


9.00598 


•9.63788 


8.34902 


44 


11.25759 


9.51189 


10.29272 


8.79669 


9.45991 


8.16858 


45 


10.99889 


9.26346 


10.07597 


8.58301 


9.27673 


7.98351 


46 


10.73562 


9.01124 


9.85425 


8.38548 


9.10007 


7.80411 


47 


10.46805 


8.75588 


9.62777 


8.14351 


8.89646 


7.60148 


48 


10.19647 


8.49745 


9.39675 


7.91815 


8.69851 


7.40385 


49 


9.92126 


8.23645 


9.16145 


7.68952 


8.49599 


7.20249 


50 


9.64271 


7.97328 


8.92214 


7.45797 


8.28906 


6.99763 


51 


9.36121 


7.70839 


8.67921 


7.22390 


8.07799 


6.78987 


52 


9.07719 


7.44225 


8.43278 


6.98770 


8.86304 


6.57929 


53 


8.79105 


7.17531 


8.18343 


6.74980 


7.64452 


6.36638 


54 


8.50325 


6.90808 


7.93147 


6.51065 


7.42277 


6.15142 


55 


8.21425 


6.64106 


7.67729 


6.27070 


7.19812 


5.93495 



JOINT LIFE ANNUITIES. 



137 



TABLE NO. XXIX. 

Annuities on Two and Three Lives, Actuaries Table- 
Makehamized. 



X 


^a 


^K 


org 


cn H 


°> H 


o B 


> 














(W 


p 


.P 


p 


P 


P 


p 


<D 
















> 


!> 


> 


i> 


> 


> 




w 


crc 


(T9 


art? 


orp 


era 




o 


ct> 


O 


o 


a> 


CD 




w 


GO 


w 


Cfi 


ID 


n 




axx 


rtXXX 


axx 


axxx 


axx 


axxx 




2 Lives 


3 Lives 


2 Lives 


3 Lives 


2 Lives 


3 Lives 


56 


7.92454 


6.37476 


7.42137 


6.03044 j 


6.97106 


5.71739 


57 


7.63462 


6.10970 j 


7.16412 


5.79036 | 


6.74165 


5.49919 


58 


7.34503 


5.84641 | 


6.90603 


5.55095 j 


6.51068 


5.28077 


59 


7.05026 


5.58542 | 


6.64759 


5.31272 j 


6.27843 


5.06266 


60 


6.76885 


5.32720 | 


6.38928 


5.07616 


6.04540 


4.85648 


61 


6.48334 


5.07231 1 


6.13164 


4.84962 


5.81207 


4.62924 


62 


6.20027 


4.82122 1 


5.87512 


4.61011 


5.57889 


4.41490 


63 


5.92016 


4.57441 | 


5.62033 


4.38158 


5.34627 


4.20278 


64 


5.64357 


4.33234 | 


5.36772 


4.15665 


5.11504 


3.99334 


65 


5.36610 


4.09544 j 


5.11782 


3.93583 


4.88537 


3.78705 


66 


5.10280 


3.86411 | 


4.87112 


3.71950 


4.65780 


3.58435 


67 


4.83966 


3.63875 | 


4.62823 


3.50807 


4.42780 


3.38564 


68 


4.58193 


3.41969 | 


4.39096 


3.30194 


4.21112 


3.19870 


69 


4.33008 


3.20723 | 


4.15739 


3.10142 


3.99293 


3.00181 


70 


4.08447 


3.00166 j 


3.92865 


2.90686 


3.77873 


2.87138 


71 


3.84552 


2.80324 j 


3.70548 


2.71840 


3.56904 


2.63836 


72 


3.61353 


2.61211 


3.48825 


2.53661 


3.36418 


2.47129 


73 


3.38883 


2.42846 | 


3.27743 


2.36131 


3.16454 


2.29765 


74 


3.17166 


2.25240 | 


3.07330 


2.19296 


2.97049 


2.13638 


75 


2.96228 


2.08401 j 


2.87632 


2.03151 


2.78233 


1.98059 


76 


2.76089 


1.92335 j 


2.68996 


1.87709 


2.60035 


1.83288 


77 


2.56767 


1.77042 j 


2.50538 


1.72978 


2.42484 


1.69087 


78 


2.38332 


1.62518 


2.33245 


1.59325 


2.25602 


1.55544 


79 


2.20128 


1.48759 


2.16882 


1.45650 


2.09414 


1.42663 


80 


2.03863 


1.35754 


1.98327 


1.33047 


1.93955 


1.30442 


81 


1.87975 


1.23494 


1.82851 


1.21144 


1.79255 


1.18879 


82 


1.73015 


1.11965 


1.68126 


1.09931 


1.65359 


1.07969 


83 


1.59046 


1.01150 


1.54508 


.99394 


1.52350 


.97698 


84 


1.40599 


.91032 


1.40955 


.89522 


1.34902 


.88061 


85 


1.30991 


.81593 


1.28459 


.80299 


1.25896 


.79044 


86 


1.11877 


.72812 


1.16659 


.71706 


1.14405 


.70617 


87 


1.07514 


.64667 


1.05842 


.63727 


1.03558 


.62813 


88 


.96882 


.57246 


.95235 


.56342 


.93267 


.55566 


89 


.86961 


.50206 


.85549 


.49532 


.83445 


.48876 


90 


.77727 


.43843 


.76525 


.43277 


.73822 


.42724 


91 


.69160 


.38029 


.68144 


.37555 


.63751 


.37093 


92 


.64120 


.32668 


.60393 


.32348 


.55164 


.31962 


93 


.53927 


.27959 


.53266 


.27635 


.48629 


.27314 


94 


.47215 


.23659 


.46824 


.23395 


.42817 


.23129 


95 


.41072 


.19188 


.41155 


.19622 


.38328 


.19386 


96 


.35472 


.16414 


.34240 


.16368 


.34613 


.16065 


97 


.30310 


.13418 


.30019 


.14100 


.29670 


.13445 


98 


.25714 


.10755 


.25416 


.11456 


.25136 


.10535 


99 


.21710 


.08598 


.20989 


.08511 


.21214 


.08622 


100 


.16710 


.06708 


.17663 


.03252 


.18117 


.05224 



138 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXX. 

Carlisle 



Annuities on Two and Three Lives 

hamized 



Table, Make- 





HH 


HH 


HH 


HH 


X 


H^ 


HH 


HH 


HH 




& 3 


•° 3 


& & 


>a & 




& ^ 


& ■£ 


>a p- 


>a tr 




tf o 


c o 


s 2 


^ 2 


P> 


£ o 


S o 


£ 2 


c <-i 




p z. 


P 7. 


p 2 


5" & 


WJ 


s° Z, 


P Z\ 


P £ 


03 CD 




H^ 


^F 


h- CD 


i— i CD 


CD 


f-» F 


£. f 


t— ■ CD 


i-< CD 


M 


t»« 


£< 


>F 


>F 




>< 


>< 


t>F 


►>F 


> 


ere cd 


ere s 


*% < 


TO 5" 




CTC CD 


CKJ CD 


ere S' 


ore 5* 


OB 


CD OQ 
QQ 


M 


02 cc 




CD CO 
03 


CD W 






axx 


axx 


axxx 


a xxx 




axx 


axx 


a xxx 


axxx 




5% 


6% 


5% 


6% 




5% 


6% 


5% 


6% 


10 


14.77563|13.01341|13.36781|11.90719 


57 


7.40845 


6.95684|6.00285|5.69040 


11 


14.67123 


12.98360 


13.24954|11.81489 


58 


7.16179 


6.73709j5.77408|5.48261 


12 


14.56368 


12.85107 


13.12781|11.71807 


59 


6.91318 


6.51495|5.54453|5.27335 


13 


14.45337 


12.76599 


13.00316111.61843 


60 


6.66396 


6.29131|5.31594|5.06437 


14 


14.34104 


12.67907 


12.87658|11.51692 


61 


6.41426 


6.066.33|5.08818|4.85525 


15 


14.22609 


12.58976 


12.74729J11.41280 


62 


6.16490 


5.84078 


4.86224 


4.64714 


16 


14.10981 


12.49917 


12.61705|11.30762 


63 


5.91605 


5.61484 


4.63820 


4.44009 


17 


13.98808 


12.40681 


12.48511 


11.20070 


64 


5.66798 


5.38876 


4.41625 


4.23428 


18 


13.87241 


12.31343 


12.35265 


11.09302 


65 


5.42153 


5.16333 


4.20693 


4.03057 


19 


13.75303 


12.21481 


12.22086 


10.98584 


66 


5.17732 


4.93915 


3.96352 


3.82954 


20 


13.63256 


12.12509 


12.08850 


10.87783 


67 


4.95379 


4.71664 


3.77032 


3.63149 


21 


13.51140 


12.02970 


11.95613 


10.76961 


68 


4.69721 


4.49614 


3.56291 


3.43671 


22 


13.38856 


11.93566 


11.82696 


10.66401 


69 


4.46209 


4.27806 


3.36002 


3.24579 


23 


13.27303 


11.84234 


11.69990 


10.56023 


70 


4.23123 


4.06325 


3.16241 


3.05924 


24 


13.15727 


11.75155 


11.57769 


10.46062 


71 


4.00454 


3.85166 


2.96985 


2.87692 


25 


13.06260 


11.67994 


11.48487 


10.38773 


72 


3.78319 


3.64436 


2.78341 


2.69992 


26 


12.96475 


11.60510 


11.38643 


10.31167 


73 


3.56696 


3.44127 


2.60276 


2.52798 


27 


12.86170 


11.52638 


11.28738 


10.23159 


74 


3.35599 


3.24028 


2.42785 


2.36107 


28 


12.75050 


11.44247 


11.18075 


10.14654 


75 


3.15123 


3.04901 


2.25958 


2.20011 


29 


12.64165 


11.35640 


11.07064 


10.05842 


76 


2.95285 


2.86112 


2.09797 


2.04517 


30 


12.52427 


11.26498 


10.95488 


9.96527 


77 


2.76083 


2.67871 


1.94285 


1.89611 


31 


12.40254 


11.16979 


10.83489 


9.86833 


78 


2.57595 


2.50261 


1.79495 


1.75368 


32 


12.27517 


11.06955 


10.70912 


9.76613 


79 


2.39720 


2.33197 


1.65299 


1.61668 


33 


12.14272 


10.96478 


10.57835 


9.65938 


80 


2.22620 


2.16830 


1.48405 


1.48672 


34 


12.00488 


10.85538 


10.44245 


9.54765 


81 


2.06210 


2.10183 


1.39068 


1.36280 


35 


11.86025 


10.74110 


10.30137 


9.43168 


82 


1.90512 


1.85988 


1.26942 


1.24511 


36 


11.71384 


10.62193 


10.15474 


9.37049 


83 


1.76809 


1.71610 


1.15540 


1.13426 


37 


11.55994 


10.49759 


10.00313 


9.18419 


84 


1.61360 


1.57866 


1.04757 


1.02925 


38 


11.40053 


10.36821 


9.84599 


9.05285 


85 


1.47871 


1.44813 


.94641 


.93057 


39 


11.23523 


10.23328 


9.68301 


8.91603 


86 


1.35128 


1.32458 


.85189 


.83825 


40 


11.06532 


10.09330 


9.51502 


8.77435 


87 


1.23089 


1.20763 


.76355 


.75184 


41 


10.88858 


9.94828 


9.34212 


8.62788 


88 


1.11722 


1.09700 


.68099 


.67098 


42 


10.70755 


9.79823 


9.16442 


8.47673 


89 


1.01060 


.99301 


.60444 


.59571 


43' 


10.52007 


9.64237 


8.98086 


8.31983 


90 


.91094 


.89556 


.53378 


.52654 


44 


10.32757 


9.48132 


8.79266 


8.15828 


91 


.81739 


.80358 


.46778 


.46165 


45 


10.12991 


9.31522 


8.59991 


7.99214 


92 


.73249 


.71972 


.40984 


.40379 


46 


9.92673 


9.14367 


8.40211 


7.82083 


93 


.65343 


.64009 


.35344 


.34915 


47 


9.71888 


8.96737 


8.20044 


7.64549 


94 


.59467 


.56873 


.30470 


.30111 


48 


9.50552 


8.78547 


7.99392 


7.46504 


95 


.50508 


.50186 


.25802 


.25507 


49 


9.28794 


8.59835 


7.78404 


7.28101 


96 


.44326 


.45142 


.22097 


.21857 


50 


9.06582 


8.4081C 


7.57048 


7.09297 


97 


.38762 


.42071 


.18739 


.18494 


51 


9.83956 


8.21268 


7.35375 


6.90137 


98 


.35331 


.38531 


.15199 


.15390 


52 


8.60876 


8.01234 


7.13319 


6.70576 


99 


.29004 


.36201 


.11081 


.10967 


53 


8.37472 


7.80834 


6.91089 


6.50746 


100 


.23722 


.34122 


.09475 


.09374 


54 


8.13755 


7.60072 


6.6865C 


6.30673 












55 


7.89680 


7.3891S 


6.4597C 


6.10304 












56 


7.65400 


7.1746* 


» 6.23212 


5.89789 













LIFE EXPECTANCY 



139 



TABLE NO. XXXI 
Expectation of Life By Tables Named 



> 

CD 
X 


North 
ampton 


o 

& 

re' 

ST 


> 

C 
P 


American 
Experience 


X 

> 
era 

CD 


e 2 

a o 

*° 2- 
<■♦■ £• 

a 


O 

P 

re" 


> 
a 

a 
p 

CD* 

re 


it 

a a 

CD 





25.18 


38.72 






53 


16.54 


18.97 


18.16 


18.79 


1 


32.74 


44.68 






54 


16.60 


18.28 


17.50 


18.09 


2 


37.79 


47.55 






55 


15.58 


17.58 


16.86 


17.40 


3 


39.55 


49.82 






56 


15.10 


16.89 


16.22 


16.72 


4 


40.58 


50.76 






57 


14.63 


16.21 


15.59 


16.05 


5 


40.84 


51.25 






58 


14.15 


15.55 


14.97 


15.39 


6 


41.07 


51.17 






59 


13.68 


14.92 


14.37 


14.74 


7 


41.03 


50.80 






60 


13.21 


14.34 


13.77 


14.10 


8 


40.79 


50.24 






61 


12.75 


13.82 


13.18 


13.47 


9 


40.36 


49.57 






62 


12.28 


13.31 


12.61 


12.86 


10 


39.78 


48.82 


48.36 


48.72 


63 


11.81 


12.81 


12.05 


12.26 


11 


39.14 


48.04 


47.68 


48.08 


64 


11.35 


12.30 


11.51 


11.67 


12 


38.49 


47.27 


47.01 


47.45 


65 


10.88 


11.79 


10.97 


11.10 


13 


37.83 


46.51 


46.33 


46.80 


66 


10.42 


11.27 


10.46 


10.54 


14 


37.17 


45.75 


45.64 


46.16 


67 


9.96 


10.75 


9.96 


10.00 


13 


36.51 


45.00 


44.96 


45.50 


68 


9.50 


10.23 


9.47 


9.47 


16 


35.85 


44.27 


44.27 


44.85 


69 


9.05 


9.70. 


9.00 


8.97 


37 


35.20 


43.57 


43.58 


44.19 


70 


8.60 


9.18 


8.54 


8.48 


18 


34.58 


42.87 


42.88 


43.53 


71 


8.17 


8.65 


8.10 


8.00 


19 


33.99 


42.17 


42.19 


42.87 


72 


7.74 


8.16 


7.67 


7.55 


20 


33.43 


41.46 


41.49 


42.20 


73 


7.33 


7.72 


7.26 


7.11 


21 


32.90 


40.75 


40.79 


41.53 


74 


6.92 


7.33 


6.86 


6.68 


22 


32.39 


40.04 


40.09 


40.85 


75 


6.54 


7.01 


6.48 


6.27 


23 


31.88 


39.31 


39.39 


40.17 


76 


6.18 


6.69 


6.11 


5.88 


24 


31.36 


38.59 


38.68 


39.49 


77 


5.83 


6.40 


5.76 


5.49 


25 


30.85 


37.86 


37.98 


38.81 


78 


5.48 


6.12 


5.42 


5.11 


26 


30.33 


37.14 


37.27 


38.12 


79 


5.11 


5.80 


. 5.09 


4.74 


27 


29.82 


36.41 


36.56 


37.43 


80 


4.75 


5.51 


4.78 


4.39 


28 


29.30 


35.69 


35.86 


36.73 


81 


4.41 


5.21 


4.48 


4.05 


29 


28.79 


35.00 


35.15 


36.03 


82 


4.09 


4.93 


4.18 


3.71 


30 


28.27 


34.34 


34.43 


35.33 


83 


3.80 


4.65 


3.90 


3.39 


31 


27.76 


33.68 


33.72 


34.63 


84 


3.58 


4.39 


3.63 


3.08 


32 


27.24 


33.03 


33.01 


33.92 


85 


3.37 


4.12 


3.36 


2.77 


33 


26.72 


32.36 


32.30 


33.21 


86 


3.19 


3.90 


3.10 


2.47 


34 


26.20 


31.68 


31.58 


32.50 


87 


3.01 


3.71 


2.84 


2.18 


35 


25.68 


31.00 


30.87 


31.78 


88 


2.86 


3.59 


2.59 


1.91 


36 


25.16 


30.32 


30.15 


31.07 


89 


2.66 


3.47 


2.35 


1.66 


37 


24.64 


29.64 


29.44 


30.35 


90 


2.41 


3.28 


2.11 


1.42 


38 


24.12 


28.96 


28.72 


29.62 


91 


2.09 


3.26 


1.89 


1.19 


39 


23.60 


28.28 


28.00 


28.90 


92 


1.75 


3.37 


1.67 


.98 


40 


23.08 


27.61 


27.28 


28.18 


93 


1.37 


3.48 


1.47 


.80 


41 


22.56 


26.97 


26.56 


27.45 


94 


1.05 


3.53 


1.28 


.64 


42 


22.04 


26.34 


25.84 


26.72 


75 


.75 


3.53 


1.12 


.50 


43 


21.54 


25.71 


25.12 


26.00 


96 


.50 


3.46 


.99 




44 


21.03 


25.09 


24.40 


25.27 


97 




3.28 


.89 




45 


20.52 


24.46 


23.69 


24.54 


98 




3.07 


.75 




46 


20.22 


23.82 


22.97 


23.81 


99 




2.77 


.50 




47 


19.51 


23.17 


22.27 


23.08 


100 




2.28 






-48 


19.00 


22.50 


21.56 


22.36 


101 




1.79 






49 


18.49 


21.81 


20.87 


21.63 


102 




1.30 






50 


1 17.99 


21.11 


20.18 


20.91 


103 




.83 






51 


17.50 


20.39 


19.50 


20.20 


104 




.50 






52 


| 17.02 


19.68 


18.82 


19.49 













140 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXXII. 

Actuaries or Combined Experience Commutation 
Columns at Three Per Cent. 



x Age 


Dx 


Nx 


Mx 


Rx 


10 


74409.391 


1737895.578 


21623.808 


743735.36 


11 


71753.769 


1666141.818 


21135.452 


722111.55 


12 


69191.125 


1596950.693 


20662.722 


700976.10 


13 


66718.250 


1530232.443 


20205.122 


680313.37 


14 


64331.391 


1465901.052 


19761.512 


660108.25 


15 


62026.971 


1403874.081 


19330.923 


640346.74 


16 


59802.215 


1344071.866 


18912.678 


621015.92 


17 


57653.833 


1286418.033 


18506.107 


602103.24 


18 


55579.278 


1230838.755 


18110.790 


583597.13 


19 


53575.521 


1177263.234 


17725.847 


565486.34 


20 


51640.230 


1125623.004 


17351.009 


547760.50 


21 


49770.613 


1075852.391 


16985.475 


530409.49 


22 


47964.530 


1027887.861 


16629.022 


513424.01 


23 


46219.915 


981667.946 


16281.432 


495794.99 


24 


44534.270 


937133.676 


15941.998 


480513.56 


25 


42905.695 


894227.981 


15610.539 


464571.56 


26 


41332.357 


852895.624 


15286.880 


448961.02 


27 


39812.019 


813083.605 


14970.398 


433674.14 


28 


38342.995 


774740.610 


14660.947 


418703.74 


29 


36923.226 


737817.384 


14357.964 


404042.80 


30 


35551.161 


702266.223 


14061.333 


389684.83 


31 


34224.900 


668041.323 


13770.543 


375623.50 


32 


32943.018 


635098.305 


13485.503 


361852.96 


33 


31703.760 


603394.545 


13205.750 


348367.45 


34 


30505.816 


572888.729 


12931.216 


335161.70 


35 


29347.917 


543540.812 


12661.835 


322230.49 


36 


28228.483 


515312.329 


12397.196 


309568.65 


37 


27146.347 


488165.982 


12137.249 


297171.46 


38 


26100.374 


462065.608 


11881.946 


285034.21 


39 


25089.146 


436976.462 


11630.922 


273152.26 


40 


24111.615 


412864.847 


11384.144 


261521.34 


41 


23166.768 


389698.079 


11141.577 


250137.19 


42 


22253.327 


367444.752 


10902.897 


238995.62 


43 


21369.797 


346074.955 


10667.521 


228092.72 


44 


20513.953 


325561.002 


10434.099 


217425.20 


45 


19683.489 


305877.513 


10201.128 


206991.10 


46 


18876.809 


287000.704 


9967.755 


196789.97 


47 


18091.700 


268909.004 


9732.454 


186822.22 


48 


17327.356 


251581.648 


9495.054 


177089.76 


49 


16582.791 


234998.857 


9255.169 


167594.71 


50 


15857.320 


219141.537 


9012.692 


158339.54 


51 


15150.075 


203991.462 


8767.311 


149326.85 


52 


14460.256 


189531.206 


8518.756 


140559.54 


53 


13787|122 


175744.084 


8266.794 


132040.78 


54 


13129.988 


162614.096 


8011.227 


123773.99 



INSURANCE COMMUTATION COLUMNS. 



141 



TABLE NO. XXXII. 

Actuaries or Combined Experience Commutation Col- 
umns at Three Per Cent -Concluded. 



x Age Dx 



Nx 



Mx 



Rx 



55 


12488.615 


150125.481 


7752.2815 


115762.7605 


56 


11862.195 


138263.286 


7489.6069 


108010.4790 


57 


11250.356 


127012.930 


7223.2693 


100520.8721 


58 


10653.112 


116359.818 


6953.7048 


93297.6028 


59 


10069.925 


106289.893 


6680.8029 


86343.8980 


60 


9500.470 


96789.423 


6404.6472 


79663.0951 


61 


8943.945 


87845.478 


6124.8348 


73258.4479 


62 


8400.260 


79445.218 


5841.6530 


67133.6131 


63 


7869.164 


71576.054 


5555.2249 


61291.9601 


64 


7350.871 


64225.283 


5266.1305 


55736.7352 


65 


6845.405 


57379.778 


4974.7681 


50470.6047 


66 


6353.056 


51026.723 


4681.7995 


45495.8366 


67 


5874.333 


45152.389 


4388.1174 


40814.0371 


68 


5409.666 


39742.723 


4094.5478 


36425.9197 


69 


4959.930 


34782.794 


3802.3741 


32331.3719 


70 


4926.118 


30256.675 


3513.0269 


28528.9978 


71 


4108.956 


26147.719 


3227.6930 


25015.9709 


72 


3709.397 


22438.322 


2947.8127 


21788.2779 


73 


3328.356 


19109.966 


2674.8128 


18840.4652 


74 


2966.814 


16143.151 


2410.2132 


16165.6524 


75 


2625.580 


13517.572 


2155.3903 


13755.4392 


76 


2305.513 


11212.058 


1911.7973 


11600.0489 


77 


2007.410 


9204.648 


1680.8446 


9688.2516 


78 


1731.695 


7472.954 


1463.5977 


8007.4070 


79 


1478.759 


5994.195 


1261.0997 


6543.8093 


80 


1248.956 


4745.240 


1074.3672 


5282.7096 


81 


1042.325 


3702.915 


904.1136 


4208.3424 


82 


858.718 


2844.197 


750.8660 


3304.2288 


83 


697.651 


2146.546 


614.8103 


2553.3628 


84 


558.180 


1588.365 


495.6595 


1938.5525 


85 


439.132 


1149.234 


392.8685 


1442.8930 


86 


338.901 


810.333 


305.4280 


1050.0245 


87 


255.827 


554.506 


232.2254 


744.5965 


88 


188.211 


366.295 


172.0602 


512.3711 


89 


134.256 


232.039 


123.5870 


340.3109 


90 


92.2347 


139.804 


85.4763 


216.7239 


91 


60.5588 


79.2455 


56.4868 


131.2476 


92 


37.5708 


41.6746 


35.2626 


74.7608 


93 


21.6939 


19.9808 


20.4801 


39.4982 


94 


11.4319 


8.5489 


10.8499 


19.0181 


95 


5.3685 


3.1804 


5.1195 


8.1682 


96 


2.1668 


1.0135 


2.0742 


3.0487 


97 


0.7392 


0.2744 


0.7096 


0.9745 


98 


0.2208 


0.0536 


0.2128 


0.2648 


99 


0.0536 


0.0000 


| 0.0520 


0.0520 



142 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXXIII. 

Actuaries or Combined Experience Commutation Col- 
umns at Three and One-Half Per Cent. 



x Age 


Dx 


Nx 


Mx 


Rx 


10 


70891.881 


1506695.178 


17543.524 


561018.40 


11 


68031.548 


1438663.630 


17080.501 


543474.87 


12 


65284.922 


1373378.708 


16634.459 


526394.37 


13 


62647.540 


1310731.168 


16204.779 


509759.91 


14 


60114.491 


1250616.677 


15790.248 


493555.13 


15 


57681.122 


1192935.555 


15389.734 


477764.89 


16 


55343.582 


1137591.973 


15002.764 


462375.13 


17 


53097.620 


1084494.353 


14628.324 


447372.39 


18 


50939.731 


1033554.622 


14266.007 


432744.06 


19 


48866.026 


984688.596 


13914.901 


418478.06 


20 


46873.313 


937815.283 


13574.664 


404563.16 


21 


44958.039 


892857.244 


13244.476 


390988.49 


22 


43117.289 


849739.955 


12924.046 


377744.02 


23 


41348.262 


808391.693 


12613.092 


364819.97 


24 


39647.821 


768743.872 


12310.S02 


352206.88 


25 


38013.408 


730730.464 


12017.238 


339895.98 


26 


36442.563 


694287.901 


11731.869 


327878.74 


27 


34932.512 


659355.389 


11454.176 


316146.87 


28 


33481.028 


625874.381 


11183.964 


304692.69 


29 


32085.513 


593788.868 


10920.678 


293508.73 


30 


30743.976 


563044.892 


10664.158 


282588.05 


31 


29454.070 


533590.822 


10413.902 


271923.89 


32 


28213.917 


505376.905 


10169.781 


261509.99 


33 


27021.387 


478355.518 


9931.345 


251340.21 


34 


25874.763 


452480.755 


9698.488 


241408.86 


35 


24772.389 


427708.366 


9471.106 


231710.38 


36 


23712.374 


403995.992 


9248.804 


222239.27 


37 


22693.202 


381302.790 


9031.499 


212990.47 


38 


21713.407 


359589.383 


8819.108 


203958.97 


39 


20771.315 


338818.058 


8611.285 


195139.86 


40 


19865.580 


318952.488 


8407.965 


186528.57 . 


41 


18994.913 


299957.575 


8209.079 


178120.61 


42 


18157.820 


281799.755 


8014.325 


169911.53 


43 


17352.658 


264447.097 


7823.196 


161897.20 


44 


16577.227 


247869.870 


7634.568 


154074.01 


45 


15829.291 


232040.579 


7447.216 


146439.44 


46 


15107.229 


216933.350 


7260.445 


138992.22 


47 


14408.955 


202524.395 


7073.043 


131731.78 


48 


13733.534 


188790.861 


6884.881 


124658.74 


49 


13079.903 


175710.958 


6695.669 


117773.86 


50 


12447.253 


163263.705 


6505.335 


111078.19 


51 


11834.648 


151429.057 


6313.653 


104572.85 


52 


11241.219 


140187.838 


6120.430 


98259.20 


53 


10666.156 


129521.682 


5925.504 


92138.77 


54 


10108.704 


119412.978 


5728.745 


86213.26 



INSURANCE COMMUTATION COLUMNS. 



143 



TABLE NO. XXXIII. 

Actuaries or Combined Experience Commutation Col- 
umns at Three and One-Half Per Cent Concluded. 



x Age 


Dx 


Nx 


Mx 


Rx 


55 


9568.466 


109844.512 


5530.3468 


80484.520 


56 


9044.613 


100799.899 


5330.0643 


74954.173 


57 


8536.663 


92263.236 


5127.9699 


69624.109 


58 


8044.429 


84218.807 


4924.4150 


64496.139 


59 


7567.316 


76651.491 


4719.3355 


59571.724 


60 


7104.894 


69546.597 


4512.8134 


54852.388 


61 


6656.385 


62890.212 


4304.5676 


50339.575 


62 


6221.555 


56668.657 


4094.8323 


46035.007 


63 


5800.049 


50868.608 


3883.7174 


41940.175 


64 


5391.860 


45476.748 


3671.6667 


38056.458 


65 


4996.846 


40479.902 


3458.9849 


34384.791 


66 


4615.050 


35864.852 


3246.1637 


30925.806 


67 


4246.677 


31618.175 


3033.8549 


27679.642 


68 


3891.866 


27726.309 


2822.6526 


24645.788 


69 


3551.075 


24175.234 


2613.4700 


21823.135 


70 


3224.832 


20950.402 


2407.3118 


19209.665 


71 


2913.463 


18036.939 


2204.9952 


16802.353 


72 


2617.449 


15419.490 


2007.5041 


14579.358 


73 


2337.229 


13082.261 


1815.7987 


12589.854 


74 


2073.285 


11008.976 


1630.8898 


10774.055 


75 


1825.959 


9183.017 


1453.6734 


9143.165 


76 


1595.621 


7587.396 


1285.0851 


7689.492 


77 


1382.596 


6204.800 


1126.0173 


6404.407 


78 


1186.937 


5017.863 


977.1121 


5278.389 


79 


1008.672 


4009.191 


838.9866 


4301.277 


80 


847.806 


3161.385 


712.2303 


3462.291 


81 


704.125 


2457.260 


597.2182 


2750.061 


82 


577.290 


1879.970 


494.1945 


2152.842 


83 


466.744 


1413.225 


403.1702 


1658.648 


84 


371.631 


1041.594 


323.8407 


1255.478 


85 


290.957 


750.637 


255.7341 


931.6369 


86 


223.462 


527.175 


198.0782 


675.9028 


87 


167.871 


359.304 


150.0435 


477.8246 


88 


122.905 


236.399 


110.7546 


327.7810 


89 


87.2478 


149.1515 


79.2537 


217.0264 


90 


59.6504 


89.5011 


54.6066 


137.7727 


91 


38.9756 


50.5255 


35.9490 


83.1661 


92 


24.0637 


26.4618 


22.3551 


47.2171 


93 


13.8276 


12.6342 


12.9328 


24.8620 


94 


7.2515 


5.3827 


6.8242 


11.9292 


95 


3.3889 


1.9939 


3.2069 


5.1050 


96 


1.3612 


0.6326 


1.2938 


1.8982 


97 


0.4621 


0.1706 


0.4407 


0.6044 


98 


0.1374 


0.0332 


0.1316 


0.1637 


99 


0.0332 


0.0000 


0.0321 


0.0321 



144 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXXIV. 

Actuaries or Combined Experience Commutation 
Columns at Four Per Cent. 



x Age 


Dx 


Nx 


Mx 


Rx 


10 


67556.41 


1314214.95 


14411.365 


427355.116 


11 


64518.98 


1249695.97 


13972.249 


412943.751 


12 


61616.50 


1188079.47 


13551.270 


398971.502 


13 


58843.05 


1129236.42 


13147.684 


385420.232 


14 


56192.37 


1073044.05 


12760.199 


372272.548 


15 


53658.54 


1019385.51 


12387.616 


359512.349 


16 


51236.50 


968149.01 


12029.364 


347124.733 


17 


48920.88 


919228.13 


11684.377 


335095.369 


18 


46707.09 


872521.04 


11352.165 


323410.992 


19 


44590.28 


827930.76 


11031.782 


312058.827 


20 


42566.30 


785364.46 


10722.808 


301027.045 


21 


40630.73 


744733.73 


10424.401 


290304.237 


22 


38779.81 


705953.92 


10136.205 


279879.836 


23 


37009.95 


668943.97 


9857.877 


269743.631 


24 


35317.31 


633626.66 


9588.693 


259885.754 


25 


33698.62 


599928.04 


9328.362 


250297.061 


26 


32150.76 


567777.28 


9076.601 


240968.699 


27 


30670.38 


537106.90 


8832.789 


231892.098 


28 


29254.64 


507852.26 


8596.687 


223059.309 


29 


27900.52 


479951.74 


8367.742 


214462.622 


30 


26605.43 


453346.31 


8145.752 


206094.880 


31 


25366.62 


427979.69 


7930.226 


197949.128 


32 


24181.75 


403797.94 


7720.993 


190018.902 


33 


23048.30 


380749.64 


7517.615 


182297.909 


34 


21964.17 


358785.47 


7319.951 


174780.293 


35 


20927.30 


337858.17 


7127.862 


167460.342 


36 


19935.51 


317922.66 


6940.969 


160332.480 


37 


18986.95 


298935.71 


6759.154 


153391.511 


38 


18079.83 


280855.88 


6582.305 


146632.357 


39 


17212.24 


263643.64 


6410.092 


140050.052 


40 


16382.56 


247261.08 


6242.419 


133639.960 


41 


15589.23 


231671.85 


6079.193 


127397.541 


42 


14830.58 


216841.27 


5920.126 


121318.438 


43 


14104.82 


202736.45 


5764.770 


115398.223 


44 


13409.74 


189326.71 


5612.184 


109633.453 


45 


12743.15 


176583.56 


5461.358 


104021.269 


46 


12103.40 


164480.16 


5311.724 


98559.911 


47 


11488.46 


152991.70 


5162.305 


93248.187 


48 


10897.30 


142094.40 


5013.002 


88085.882 


49 


10328.76 


131765.64 


4863.588 


83072.880 


50 


9781.92 


121983.72 


4714.010 


78209.292 


51 


9255.78 


112727.94 


4564.097 


73495.280 


52 


8749.39 


103978.54 


4413.706 


68931.184 


53 


8261.89 


95716.65 


4262.718 


64517.480 


54 


7792.45 


87924.20 


4111.043 


60254.760 



INSURANCE COMMUTATION COLUMNS. 



145 



TABLE NO. XXXIV. 

Actuaries or Combined Experience Commutation 
Columns at Four Per-Cent — Concluded. 



x Age 



Dx 



Xx 



Mx 



Rx 



55 


7340.540 


80583.661 


3958.840 


56143.718 


56 


6905.301 


73678.360 


3805.930 


52184.877 


57 


6486.161 


67192.179 


3652.379 


48378.947 


58 


6082.776 


61109.403 


3498.461 


44726.567 


59 


5694.498 


55414.905 


3344.137 


41228.106 


60 


5320.816 


50094.089 


3089.473 


37883.970 


61 


4960.965 


45133.124 


3034.269 


34694.497 


62 


4614.595 


40518.529 


2S78.706 


31660.228 


63 


4281.277 


36237.252 


2722.872 


28781.522 


64 


3960.841 


32276.411 


2567.101 


26058.650 


65 


3653.017 


28623.394 


2411.617 


23491.549 


66 


3357.679 


25265.715 


2256.779 


21079.932 


67 


3074.814 


22190.901 


2103.056 


18823.154 


68 


2804.366 


19386.535 


1950.870 


16720.098 


69 


2546.500 


16840.035 


1800.864 


14769.228 


70 


2301.431 


1453S.604 


1653.737 


12968.364 


71 


2069.223 


12469.381 


1510.046 


11314.627 


72 


1850.048 


10619.333 


1370.457 


9804.581 


73 


1644.044 


8975.289 


1235.608 


8434.125 


74 


1451.369 


7523.920 


1106.166 


7198.517 


75 


1272.086 


6251.834 


982.7048 


6092.3507 


76 


1106.275 


5145.559 


865.8194 


5109.6461 


77 


953.971 


4191.588 


756.0649 


4243.8267 


78 


815.031 


3376.557 


653.8165 


3487.7618 


79 


689.294 


2687.263 


559.4261 


2833.9453 


80 


576.578 


2110.685 


473.2214 


2274.5192 


81 


476.560 


1634.125 


395.3799 


1801.2978 


82 


388.838 


1245.287 


325.9874 


1405.9179 


83 


312.868 


932.419 


264.9721 


1079.9304 


84 


247.914 


684.505 


212.05.16 


814.9583 


85 


193.163 


491.342 


166.8363 


602.9067 


86 


147.640 


343.702 


128.7432 


436.0704 


87 


110.379 


233.3228 


97.1593 


307.3272 


88 


80.4242 


152.8986 


71.4502 


210.1678 


89 


56.8171 


96.0815 


50.9363 


138.7176 


90 


38.6584 


57.4231 


349630 


87.7813 


91 


25.1380 


32.2851 


22.9294 


52.8183 


92 


15.4457 


16.8394 


14.2040 


29.8889 


93 


8.8328 


8.0066 


8.1851 


15.6849 


94 


4.6098 


3.3968 


4.3019 


7.4998 


95 


2.1440 


1.2528 


2.0133 


3.1979 


96 


.8570 


.3958 


.8089 


1.1846 


97 


.2895 


.1063 


.2743 


.3757 


98 


.0857 


.0206 


.0816 


.1014 


99 


.0206 


.0000 


.0198 


.0198 



146 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXXV. 

American Experience Commutation Columns, 
Three Per Cent. 



x Age 



Dx 



Nx 



Mx 



Rx 



10 


74409.40 


1736936.32 


21651.7690 


734216.0588 


11 


71701,05 


1665235.27 


21110.6754 


712564.2898 


12 


69089.43 


1596145.84 


20587.4460 


691453.6144 


13 


66571.17 


1529574.67 


20081.4991 


670866.1684 


14 


64142.87 


1465431.80 


19592.2719 


650784.6693 


15 


61801.66 


1403630.14 


19119.2196 


631192.3974 


16 


59543.60 


1344086.54 


18661.1719 


612073.1778 


17 


57366.45 


1286720.09 


18218.2999 


593412.0059 


18 


55267.37 


1231452.72 


17790.0893 


575193.7060 


19 


53243.04 


1178209.68 


17375.4904 


557403.6167 


20 


51290.86 


1126918.82 


16974.0755 


540028.1263 


21 


49408.31 


1077510.51 


16585.4274 


523054.0508 


22 


47592.42 


1029918.09 


16208.6209 


506468.6234 


23 


45840.91 


984077.18 


15843.2961 


490260.0025 


24 


44151.55 


939925,63 


15489.1038 


474416.7064 


25 


42522.27 


897403.36 


15145.7054 


458927.6026 


26 


40950.73 


856452.63 


14812.7728 


443781.8972 


27 


39434.76 


817017.87 


14489.5371 


428969.1244 


28 


37972,35 


779045.52 


14175.7160 


414479.5873 


29 


36561.69 


742483.83 


13871.0353 


400303.8713 


30 


35200.56 


707283.27 


13574.8168 


386432.8360 


31 


33887.31 


673395.96 


13286.8261 


372858.0192 


32 


32620.32 


640775.44 


13006.8350 


359571.1931 


33 


31397.61 


609377.83 


12734.2450 


346564.3581 


34 


30217.37 


579160.46 


12468.4965 


333830.1131 


35 


29078.18 


550082.28 


12209.4220 


321361.6166 


36 


27978.68 


522103.60 


11956.8583 


309152.1946 


37 


26916.89 


495186.71 


11709.9759 


297195.3363 


38 


25891.58 


469295.13 


11468.6581 


285485.3604 


39 


24900.82 


444394.31 


11232.1587 


274016.7023 


40 


23943.93 


420450.38 


11000.4017 


262784.5436 


41 


23018.85 


397431.53 


10772.7163 


251784.1419 


42 


22124.74 


375306.79 


10549.0619 


241011.4256 


43 


21260.11 


354046.68 


10328.8357 


230462.3637 


44 


20423.80 


333622.88 


10111.7554 


220133.5280 


45 


19614.20 


314008.68 


9897.0318 


210021.7726 


46 


18830.34 


295178.34 


9684.4540 


200124.7408 


47 


18070.51 


| 277107.83 


9473.0825 


190440.2868 


48 


17333.65 


1 259774.18 


9262.5435 


180967.2043 


49 


16618.26 


1 243155.92 


9052.0281 


171704.6608 


50 


15922.79 


| 227233.13 


8840.5728 


162652.6327 


51 


15245.97 


211987.16 


8627.5252 


153812.0599 


52 


14586.69 


197400.47 


8412.2973 


145184.5347 



INSURANCE COMMUTATION COLUMNS. 



147 



TABLE NO. XXXV. 

American Experience Commutation Columns, Three 
Per Cent — Concluded. 



x Age 


Dx 


Nx 


Mx 


Rx 


53 


13943.89 


183456.58 


8194.3620 


136772.2374 


54 


13316.65 


170139.93 


7973.2488 


128577.8754 


55 


12703.88 


157436.05 


7748.3440 


120604.6266 


56 


12104.71 


145331.34 


7519.2917 


112856.2826 


57 


11518.55 


133812.79 


7285.5971 


105336.9909 


58 


10944.46 


122868.33 


1047.0045 


98051.3938 


59 


10382.00 


112486.33 


6803.2983 


91004.3893 


60 


9830.43 


102655.90 


6554.1301 


84201.0910 


61 


9289.34 


93366.56 


6299.3656 


77646.9609 


62 


8758.32 


84608.24 


6038.9024 


71347.5953 


63 


8237.14 


76371.10 


5772.8224 


65308.6929 


64 


7727.78 


68645.32 


5501.3722 


59535.8705 


65 


7224.18 


61421.14 


5224.7976 


54034.4983 


66 


6732.31 


54688.33 


4943.3430 


48809.7007 


67 


6250.54 


48438.29 


4657.6654 


43866.3577 


68 


5779.34 


42658.95 


4368.5175 


39208.6923 


69 


5319.23 


37339.72 


4076.7341 


34840.1748 


70 


4871.16 


32468.56 


3783.5980 


30763.4407 


71 


4436.10 


28032.46 


3490.4164 


26979.8427 


72 


4015.47 


24016.99 


3198.9885 


23489.4363 


73 


3611.06 


20405.93 


2911.5411 


20290.4478 


74 


3224.79 


17181.14 


2630.4461 


17378.9067 


75 


2858.39 


14322.75 


2357.9742 


14748.4606 


76 


2513.25 


11809.50 


2096.0826 


12390.4864 


77 


2190.41 


9619.09 


1846.4400 


10294.4038 


78 


1890.42 


7728.67 


1610.2499 


8447.9638 


79 


1613.60 


6115.07 


1388.4893 


6837.7139 


80 


1360.22 


4754.85 


1182.1156 


5449.2246 


81 


1129.82 


3625.026 


991.3330 


4267.1090 


82 


922.940 


2702.086 


817.3571 


3275.7760 


83 


739.878 


1962.208 


661.1768 


2458.4189 


84 


580.725 


1381.483 


523.5731 


1797.2421 


85 


444.644 


936.839 


404.4069 


1273.6690 


86 


330.007 


606.832 


302.7209 


869.2621 


87 


235.274 


371.558 


217.5979 


566.5412 


88 


159.204 


212.354 


148.3820 


348.9433 


89 


100.980 


111.374 


94.7949 


200.5613 


90 


59.229 


52.145 


55.9850 


105.7664 


91 


31.366 


20.779 


29.8470 


49.7814 


92 


14.237 


6.452 


13.6322 


19.9344 


93 


5.056 


1.486 


4.8650 


6.3022 


94 


1.305 


.181 


1.2615 


1.4372 


95 


.181 


.000 


0.1757 


0.1757 



148 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXXVI. 

American Experience Commutation Columns, at 
Three and One-Half Per Cent. 



xAge 


Dx 


Nx 


Mx 


Rx 


10 


70891.89 


1504643.35 


17612.9031 


552495.7869 


11 


67981.56 


1436661.79 


17099.8787 


534882.8838 


12 


65188.97 


1371472.82 


16606.1884 


517783.0051 


13 


62509.35 


1308963.47 


16131.1112 


501176.8167 


14 


59938.42 


1249025.05 


15673.9527 


485045.7055 


15 


57471.61 


1191553.44 


15234.0442 


469371.7528 


16 


55104.25 


1136449.19 


14810.1742 


454137.7086 


17 


52832.94 


1083616.25 


14402.3011 


439327.5344 


18 


50653.87 


1032962.38 


14009.8359 


424925.2333 


19 


48562.77 


984399.61 


13631.6827 


410915.3974 


20 


46556.19 


937843.42 


13267.3224 


397283.7147 


21 


44630.76 


893212.66 


12916.2547 


384016.3923 


22 


42782.78 


850429.88 


12577.5278 


371100.1376 


23 


41009.20 


809420.68 


12250.7089 


358522.6098 


24 


39307.09 


770113.59 


11935.3798 


346271.9009 


25 


37673.62 


732439.97 


11631.1371 


334336.5211 


26 


36106.08 


696333.89 


11337.5917 


322705.3840 


27 


34601.50 


661732.39 


11053.9729 


311367.7923 


28 


33157.37 


628575.02 


10779.9451 


300313.8194 


29 


31771.35 


596803.67 


10515.1840 


289533.8743 


30 


30440.78 


566362.89 


10259.0198 


279018.6903 


31 


29163.54 


537199.35 


10011.1740 


268759.6715 


32 


27937.53 


509261.82 


9771.3768 


258748.4975 


33 


26760.46 


482501.36 


9539.0460 


248977.1207 


34 


25630.11 


456871.25 


9313.6404 


239438.0747 


35 


24544.71 


432326.54 


9094.9573 


230124.4343 


36 


23502.54 


408824.00 


8882.7998 


221029.4770 


37 


22501.38 


386322.62 


8676.4165 


212146.6772 


38 


21539.71 


364782.91 


8475.6595 


203470.2607 


39 


20615.52 


344167.39 


8279.8615 


194994.6012 


40 


19727.43 


324439.96 


8088.9171 


186714.7397 


41 


18873.63 


305566.33 


7902.2332 


178625.8226 


42 


18052.89 


287513.44 


7719.7402 


170723.5894 


43 


17263.59 


270249.85 


7540.9125 


163003.8492 


44 


16504.41 


253745.44 


7365.4910 


155462.9367 


45 


15773.57 


237971.87 


7192.8117 


148097.4457 


46 


15070.02 


222901.85 


7022.6843 


140904.6340 


47 


14392.08 


208509.77 


6854.3396 


133881.9497 


48 


13738.52 


194771.25 


6687.4680 


127027.6101 


49 


13107.89 


181663.36 


6521.4211 


120340.1421 


50 


12498.64 


169164.72 


6355.4387 


113818.7210 


51 


11909.54 


157255.18 


6189.0142 


107463.2823 


52 


11339.50 


145915.68 


6021.6988 


101274.2681 



INSURANCE COMMUTATION COLUMNS. 



149 



TABLE NO. XXXVI. 

American Experience Commutation Columns at 
Three and One-Half Per Cent— Concluded. 



x Age 


Dx 


Nx 


Mx 


Rx 


53 


| 10787.47 


135128.21 


5853.0971 


95252.5693 


54 


| 10252.41 


124875.796 


5682.8633 


89399.4722 


55 


9733.398 


115142.398 


5510.5468 


83716.6089 


56 


9229.602 


105912.796 


5335.9005 


78206.0621 


57 


1 8740.164 


97172.632 


5158.5732 


72870.1616 


58 


8264.438 


88908.194 


4978.4061 


67711.5884 


59 


7801.823 


81106.371 


4795.2666 


62733.1823 


60 


7351.656 


73754.715 


4608.9270 


57937.9157 


61 


6913.441 


66841.274 


4419.3227 


53328.9887 


62 


6486.746 


60354.528 


4226.4136 


48909.6660 


63 


6071.271 


54283.257 


4030.2965 


44683.2524 


64 


5666.851 


i 48616.406 


3831.1879 


40652.9559 


65 


5273.332 


43343.074 


3629.3005 


36821.7680 


66 


4890.551 


38452.523 


3424.8435 


33192.4675 


67 


4518.648 


33933.875 


3218.3212 


29767.6240 


68 


4157.822 


29776.053 


3010.3000 


26549.3028 


69 


3808.316 


25967.737 


2801.3968 


23539.0028 


70 


3470.674 


22497.063 


2592.5388 


20737.6060 


71 


3145.429 


19351.634 


2384.6578 


18145.0672 


72 


2833.421 


16518.213 


2179.0185 


15760.4094 


73 


2535.754 


13982.459 


1977.1678 


13581.3909 


74 


2253.567 


11728.892 


1780.7314 


11604.2231 


75 


1987.870 


9741.022 


1591.2404 


9823.4917 


76 


1739.394 


8001.628 


1409.9869 


8232.2513 


77 


1508.634 


6492.994 


1238.0465 


6822.2644 


78 


1295.725 


5197.269 


1076.1584 


5584.2179 


79 


1100.647 


4096.622 


924.8937 


4508.0595 


80 


923.338 


3173.285 


784.8046 


3583.1658 


81 


763.234 


2410.051 


655.9245 


2798.3612 


82 


620.465 


1789.586 


538.9657 


2142.4367 


83 


494.994 


1294.592 


434.4776 


1603.4710 


84 


386.641 


907.951 


342.8623 


1168.9934 


85 


294.610 


613.341 


263.9059 


826.1311 


86 


217.598 


395.744 


196.8569 


562.2252 


87 


154.383 


241.361 


141.0003 


365.3683 


88 


103.963 


137.398 


95.8011 


224.3680 


89 


65.623 


71.775 


60.9768 


128.5669 


90 


38.305 


33.470 


35.8775 


67.5901 


91 


20.187 


13.283 


19.0551 1 


31.7126 


92 


9.119 


4.164 


8.6697 1 


12.6575 


93 


3.222 


.942 


3.0816 | 


3.9878 


94 


.828 


.114 


.7958 1 


.9062 


95 


.114 


.000 


.1104 1 


.1104 



150 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXXVII. 

American Experience Commutation 
Columns at Four Per Cent. 



x Age 


Dx 


Nx 


Mx 


Rx 


10 


67556.41 


| 1311526.80 


14514.759 


420042.335 


11 


64471.56 


| 1247055.24 


14028.223 


405527.576 


12 


61525.93 


| 1185529.31 


1S562.274 


391499.353 


13 


58713.33 


| 1126815.98 


1S116.047 


377937.079 


14 


56027.78 


1070788.20 


12688.716 


364821.032 


15 


53463.64 


1017324.56 


12279.486 


352132.316 


16 


51014.92 


966309.64 


11887.063 


339852.830 


17 


48677.02 


917632.62 


11511.274 


327965.767 


18 


46444.97 


871187.65 


11151.419 


316454.493 


19 


44313.56 


826874.09 


10806.354 


305303.074 


29 


42278.32 


784595.77 


10475.473 


294496.720 


21 


40334.96 


744260.81 


10158.196 


284021.247 


22 


38478.94 


705871.87 


9853.544 


273863.051 


23 


36706.47 


669075.40 


9561.015 


264009.507 


24 


35013.80 


634061.60 


9280.128 


254448.492 


25 


33397.39 


600664.21 


9010.419 


245168.364 


26 


31853.91 


568810.30 


8751.444 


236157.945 


27 


30379.74 


538430.56 


8502.430 


227406.501 


28 


28971.85 


509458.71 


8262.993 


218904.071 


29 


27627.33 


481831.38 


8032.765 


210641.078 


30 


26343.05 


455488.33 


7811.084 


202608.313 


31 


25116.41 


430371.92 


7597.633 


194797.229 


32 


23944.87 


406427.05 


7392.106 


187199.596 


33 


22825.74 


383601.31 


7193.936 


179807.490 


34 


21756.49 


361844.82 


7002.597 


172613.554 


35 


20734.96 


341109.86 


6817.857 


165610.957 


36 


19759.10 


321350.76 


6639.491 


158793.100 


37 


18826.45 


302524.31 


6466.814 


152153.609 


38 


17935.20 


284589.11 


6299.652 


145686.795 


39 


17083.14 


267505.97 


6137.403 


139387.143 


40 


16268.62 


251237.35 


5979.936 


133249.740 


41 


15489.70 


235747.65 


5826.723 


127269.804 


42 


14744.92 


221002.73 


5677.670 


121443.081 


43 


14032.42 


206970.31 


5532.313 


115765.411 


44 


13350.81 


193619.50 


5390.410 


110233.098 


45 


12698.30 


180921.20 


5251.397 


104842.688 


46 


12073.60 


168847.60 


5115.097 


99591.291 


47 


11475.01 


157372.59 


4980.873 


94476.194 


48 


10901.26 


146471.33 


4848.464 


89495.321 


49 


10350.85 


136120.48 


4737.342 


84646.857 


50 


9822.30 


126298.17 


4586.901 


79929.515 


51 


9314.36 


116983.81 


4456.742 


75342.614 


52 


8825.89 


108157.92 


4326.515 


70885.872 



INSURANCE COMMUTATION COLUMNS. 



151 



TABLE NO. XXXVII. 

American Experience Commutation 

Columns at Four Per Cent — 

Concluded. 



x Age 


Dx 


Xx 


Mx 


Rx 


53 


8355.838 


99802.079 


| 4195.918 


66559.357 


54 


7903.230 


91898.849 


1 4064.691 


62363.439 


55 


7467.065 


84431.784 


1 3932.497 


58298.748 


56 


7046.535 


77385.249 


3799.160 


64366.251 


57 


6640.873 


70744.466 


3664.428 


50567.091 


58 


6249.134 


64495.332 


3528.195 


46902.663 


59 


5870.969 


58624.363 


3390.380 


43374.468 


60 


5505.613 


53118.750 


3250.831 


39984.088 


61 


5152.547 


47966.203 


3109.520 


36733.257 


62 


4811.291 


43154.912 


2966.437 


33623.737 


63 


4481.477 


38673.435 


2821.674 


30657.300 


64 


4162.850 


34510.585 


2675.410 


27835.626 


65 


3855.146 


30655.439 


2527.817 


25160.216 


66 


3558.120 


27097.319 


2379.064 


22632.399 


67 


3271.735 


23825.584 


2229.531 


20253.335 


68 


2996.005 


20829.579 


2079.637 


18023.804 


69 


2730.969 


18098.610 


1929.831 


15944.167 


70 


2476.878 


15621.732 


1780.778 


14014.336 


71 


2233.970 


13387.762 


1633.135 


12233.558 


72 


2002.700 


11385.062 


1487.786 


10600.423 


73 


1783.688 


9601.374 


1345.801 


9112.637 


74 


1577.573 


8023.801 


1208.289 


7766.836 


75 


1384.886 


6628.915 


1076.277 


6558.547 


76 


1205.954 


5432.961 


950.611 


5482.270 


77 


1040.936 


4392.025 


831.975 


4531.659 


78 


889.735 


3502.290 


720.811 


3699.684 


79 


752.145 


2750.145 


617.442 


2978.873 


80 


627.945 


2122.200 


522.170 


2361.431 


81 


516.566 


1605.634 


434.942 


1839.261 


82 


417.919 


1187.715 


356.164 


1404.319 


83 


331.805 


855.910 


286.124 


1048.155 


84 


257.927 


597.983 


225.007 


762.031 


85 


195.588 


402.395 


172.589 


537.024 


86 


143.767 


258.629 


128.290 


364.435 


87 


101.510 


157.118 


91.563 


236.145 


88 


68.029 


89.089 


61.986 


144.582 


89 


42.735 1 


46.354 


39.308 


82.596 


90 


24.825 


21.530 


23.042 


43.288 


91 


13.020 | 


8.510 


12.192 


20.246 


92 


5.853 


2.657 


5.526 


8.055 


93 


2.058 | 


0.598 


1.956 


2.529 


94 


0.526 | 


0.072 


0.503 


0.573 


95 


0.072 I 


0.000 


0.069 


0.069 



152 FINANCE AND LIFE INSURANCE. 

TABLE NO. XXXVILL 

American Experience Commutation 
Columns at Four and One- 
Half Per Cent. 



x Age Dx 



Nx 



Mx 



Rx 



10 


64392.77 


1149751.31 


12109.051 


322708.032 


11 


61158.34 


1088592.97 


11647.519 


310598.980 


12 


58084.84 


1030508.13 


11207.629 


298951.462 


13 


55164.32 


975343.81 


10788.375 


287743.833 


14 


52389.25 


922954.56 


10388.796 


276955.457 


15 


49752.43 


873202.13 


10007.973 


266566.661 


16 


47246.54 


825955.59 


9644.538 


256558.689 


17 


44865.64 


781089.95 


9298.173 


246914.152 


18 


42603.53 


738486.42 


8968.081 


237615.979 


19 


38411.27 


659621.23 


8352.455 


219994.827 


20 


40453.92 


698032.50 


8653.071 


228647.898 


21 


36470.32 


623150.91 


8065.577 


211642.373 


22 


34625.68 


588525.23 


7791.433 


203576.796 


23 


32872.65 


555652.58 


7529.458 


195785.363 


24 


31206.73 


524445.85 


7279.111 


188255.905 


25 


29623.67 


494822.18 


7039.878 


180976.794 


26 


28119.40 


466702.78 


6811.265 


173936.916 


27 


26689.75 


440013.03 


6592.497 


167125.651 


28 


25331.08 


414681.95 


6383.149 


160533.154 


29 


24039.93 


390642.02 


6182.816 


154150.006 


30 


22812.75 


367829.27 


5990.843 


147967.189 


31 


21646.42 


346182.85 


5806.881 


141976.347 


32 


20537.99 


325644.86 


5630.597 


136169.465 


33 


19484.42 


306160.44 


5461.436 


130538.867 


34 


18482.83 


287677.61 


5298.887 


125077.431 


35 


17530.73 


270146.88 


5142.696 


119778.544 


36 


16625.73 


253521.15 


4992.615 


114635.848 


37 


15765.20 


237755.95 


4848.016 


109643.233 


38 


14947.00 


222808.95 


4708.706 


104795.216 


39 


14168.78 


208640.17 


4574.136 


100086.511 


40 


13428.66 


195211.51 


4444.158 


95512.379 


41 


12724.53 


182486.98 


4318.296 


91068.216 


42 


12054.73 


170432.25 


4196.438 


86749.920 


43 


11417.36 


159014.89 


4078.169 


82553.483 


44 


10810.79 


148294.10 


3963.263 


78475.314 


45 


10233.23 


137970.87 


3851.236 


74512.050 


46 


9683.25 


128287.62 


3741.921 


70660.814 


47 


9159.13 


119128.49 


3634.786 


66918.893 


48 


8659.54 


110468.95 


3529.605 


63284.107 


49 


8182.981 


102285.967 


3425.946 


59754.502 


50 


7727.976 


94557.992 


3323.318 


56328.556 


51 


7293.276 


87264.716 


3221.401 


53005.239 


52 


6877.730 


80386.986 


3119.920 | 


49783.837 



INSURANCE COMMUTATION COLUMNS. 



153 



TABLE NO. XXXVIII. 

American Experience Commutation 
Columns at Four and One 
Half Per Cent— Concluded. 



x Age 



Dx 



Xx 



Mx 



Rx 



53 


6480.277 


73906.709 


3018.637 


46663.918 


54 


6099.937 


67806.772 


2917.352 


43645.281 


55 


5735.717 


62071.055 


2815.808 


40727.929 


56 


5386.794 


56684.261 


2713.877 


37912.121 


57 


5052.322 


51631.939 


2611.737 


35198.243 


58 


4731.609 


46900.330 


2508.223 


32586.870 


59 


4424.005 


42476.325 


2404.374 


30078.647 


60 


4128.846 


38347.479 


2299.722 


27674.274 


61 


3845.582 


34501.897 


2194.255 


25374.552 


62 


3573.704 


30928.193 


2087.977 


23180.297 


63 


3312.801 


27615.392 


1980.965 


21092.320 


64 


3062.540 


24552.852 


1873.361 


19111.356 


65 


2822.598 


21730.254 


1765.299 


17237.995 


66 


2592.661 


19137.593 


1656.908 


15472.697 


67 


2372.578 


16705.015 


1548.471 


13815.788 


68 


2162.230 


14602.785 


1440.292 


12267.317 


69 


1961.522 


12641.2'63 


1332.694 


10827.025 


70 


1770.509 


10870.754 


1226.148 


9494.332 


71 


1589.234 


9281.520 


1121.116 


8268.184 


72 


1417.893 


7863.626 


1018.210 


7147.068 


73 


1256.792 


6606.834 


918.167 


6128.858 


74 


1106.244 


5500.590 


821.739 


5210.691 


75 


966.479 


4534.111 


729.611 


4388.951 


76 


837.581 


3696.530 


642.332 


3659.339 


77 


719.510 


2977.021 


560.329 


3017.008 


78 


612.055 


2364.966 


483.858 


2456.679 


79 


514.931 


1850.035 


413.090 


1972.821 


80 


427.844 


1422.191 


348.177 


1559.731 


81 


350.273 


1071.919 


289.030 


1211.554 


82 


282.027 


789.892 


235.868 


922.524 


83 


222.842 


567.049 


188.828 


686.656 


84 


172.397 


394.653 


147.978 


497.828 


85 


130.104 


264.548 


113.110 


349.850 


86 


95.175 


169.373 


83.783 


236.740 


87 


66.879 


102.494 


59.586 


152.957 


88 


441606 


"571887" 


40.193 


93.371 


89 


27.887 


30.001 


25.394 


53.178 


90 


16.122 


13.879 


14.830 


27.784 


91 


8.415 


5.464 


7.817 


12.954 


92 


3.765 


1.699 


3.530 


5.137 


93 


1.318 


0.381 


1.245 


1.607 


94 


0.335 


0.046 


0.319 


0.363 


95 


0.046 


0.000 


0.044 


0.044 



154 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XXXIX. 

National Fraternal Congress Table 
Commutation Columns at 
Four Per Cent. 



x Age 


Dx 


Nx 


Sx 


Mx 


Rx 


20 


45638.69 


889569.75 


15263684.02 


9669.139 


312174.121 


21 


43663.94 


845905.81 


14374114.27 


9449.722 


3025.04.982 


22 


41773.16 . 


804132.65 


13528208.46 


9238.323 


293055.260 


23 


399*2.83 


764169.82 


12724075.81 


9034.648 


283816.938 


24 


38229.56 


725940.26 


11959905.99 


8838.417 


274782.290 


25 


36569.76 


689370.50 


11233965.73 


8648.983 


265943.873 


26 


34980.36 


654390.13 


10544595.26 


8466.113 


257294.890 


27 


33458.09 


620932.04 


9890205.11 


8289.237 


248828.777 


28 


32000.16 


588931.88 


9269273.06 


8118.163 


240539.540 


29 


30603.61 


558328.27 


8680341.18 


7952.386 


232421.377 


30 


29265.61 


529062.66 


8122012.92 


7791.444 


224468.991 


31 


27983.77 


501078.89 


7592950.26 


7635.209 


216677.547 


32 


26755.54 


474323.35 


7091871.37 


7483.273 


209042.337 


33 


25578.47 


448744.88 


6617548.02 


7335.263 


201559.064 


34 


24450.25 


424294.62 


6168803.15 


7190.836 


194223.803 


35 


23368.71 


400925.92 


5744508.52 


7049.684 


187032.965 


36 


22331.75 


378594.17 


5343582.60 


6911. 523 


179983.282 


37 


21337.41 


357256.75 


4964988.44 


6776.100 


173071.758 


38 


20383.60 


336873.15 


4607731.69 


6642.956 


166295.658 


39 


19468.34 


317404.81 


4270858.53 


6511.684 


159652.702 


40 


18590.01 


298814.80 


3953453.73 


6382.128 


153141.018 


41 


17746.83 


281067.97 


3654638.92 


6253.950 


146758.890 


42 


16937.16 


264130.82 


3373570.95 


6126.851 


140504.939 


43 


16159.26 


247971.56 


3109440.13 


6000.381 


134378.089 


44 


15411.69 


232559.87 


2861468.57 


5874.324 


128377.707 


45 


14693.27 


217866.59 


2628908.71 


5748.665 


122503.383 


46 


14002.88 


203863.71 


2411042.11 


5623.393 


116754.719 


47 


13339.26 


190524.45 


2207178.40 


5498.350 


111131.325 


48 


12701.11 


177823.34 


2016653.95 


5373.246 


105632.975 


49 


12087.19 


165736.15 


1838830.61 


5247.832 


100259.729 


50 


11496.50 


154239.65 


1673094.46 


5122.035 


95011.897 


51 


10927.82 


143311.83 


1518854.81 


4995.529 


89889.862 


52 


10379.90 


132931.93 


1375542.98 


4868.904 


84894.333 


53 


9851.950 


123079.98 


1242611.05 


4739.183 


80026.429 


54 


9342.763 
8851.579 


113737.22 


1119531.07 


4608.918 


75287.246 


55 


104885.64 


1005793.86 


4477.071 


70678.328 


56 


8377.463 


96508.174 


900908.221 


4343.400 


66201.258 


57 


7919.451 


88588.722 


804400.047 


4207.599 


61857.858 


58 


7476.876 


81111.847 


715811.324 


4069.618 


57650.259 


59 


7049.117 


74062.730 


634699.478 


3929.430 


53580.642 



INSURANCE COMMUTATION COLUMNS. 



155 



TABLE NO. XXXIX. 

National Fraternal Congress Table 
Commutation Columns at Four Per Cent — Concluded. 



x Age 


Dx 


Nx 


Sx 


Mx 


Rx 


60 


6635.311 


67427.419 


560636.748 


3786.745 


49651.212 


1 


6234.957 


61192.462 


493209.329 


3641.595 


45864.467' 


2 


5847.410 


55345.052 


432016.867 


3493.854 


42222.873 


3 


5472.! 


49872.798 


376671.815 


3343.598 


38729.019 


4 


5109.017 


44763.781 


326799.016 


3190.833 


35385.421 


5 


4757.423 


40006. 35S 


282035.235 


3035.739 


32194.588 


6 


4417.128 


35589.230 


242028.878 


2878.422 


29158.849 


7 


4087.881 


31501.349 


206439.648 


2719.065 


26280.427 


8 


3769.647 


27731.701 


174938.300 


2558.057 


23561.362 


9 


3462.366 


24269.335 


147206.599 


2395.762 


21003.305 


70 


3166.145 


21103.190 


122937.264 


2232.709 


18607.543 


1 


2881.043 


18222.147 


101834.074 


2069.382 


16374.834 


2 


2607.310 


15614.837 


83611.927 


1906.458 


14305.452 


3 


2345.348 


13269.489 


67997.090 


1744.777 


12398.994 


4 


2095.452 


11174.037 


54727.601 


1585.087 


10654.217 


5 


1858.143 


9315.893 


43553.564 


1428.373 


9069.129 


6 


1633.959 


7681.934 


34237.671 


1275.655 


7640.756 


7 


1423.441 


6258.493 


26555.736 


1127.982 


6365.1010 


8 


1227.169 


5031.324 


20297.243 


986.4579 


5237.1188 


9 


1045.649 


3985.675 


15265.919 


852.1366 


4250.6609 


80 


879.4003 


3106.2748 


11280.244 


726.1051 


3398.5243 


1 


879.4003 


3106.2748 


11280.244 


726.1051 


3398.5243 


2 


594.1278 


1783.3320 


5796.5093 


502.6870 


2063.0764 


3 


475.4340 


1307.8980 


4013.1773 


406.8443 


1560.3893 


4 


372.5940 


935.3040 


2705.2793 


322.2903 


1153.5450 


5 


285.1631 


650.1409 


1769.9754 


249.1898 


831.2547 


6 


212.4782 


437.6627 


1119.8345 


187.4728 


582.0649 


7 


153.5674 


284.0954 


682.1718 


136.7342 


394.5922 


8 


107.1794 


176.9160 


398.0764 


96.2527 


257.8580 


9 


71.8748 


105.0412 


221.1605 


65.0703 


161.6053 


90 


46.0150 


59.0262 . 


116.1193 


41.9749 


96.5350 


1 


27.9280 


31.0982 


57.0931 


25.6578 


54.5601 


2 


15.9064 


15.1918 


25.9949 


14.7103 


28.9023 


3 


8.4159 


6.7759 


10.8031 


7.8316 


14.1920 


4 


4.0586 


2.7173 


4.0272 


3.7980 


6.3604 


5 


" 1.7586 


.9587 


1.3099 


1.6540 


2.5624 


6 


.6717 


.2870 


.3512 


.6349 


.9083 


7 


.2227 


.0643 


.0643 


.2117 


.2735 


8 


.0643 


.0000 


.0000 


.0618 


.0618 



156 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XL. 

Life Annuities by Actuaries or Combined Experience at the Rates 

Stated. 



x Age 


3% 


3y 2 % 


4% 


5% 


6% 


t Age 3 % 


8% % 


4% 


5% 


6% 


10 


23.256 


21.253 


19.454 


16.556 


14.347 


55 


12.021 


11.479 


10.978 


10.077 


9.295 


11 


23.220 


21.149 


19.369 


16.502 


14.312 


56 


11.656 


11.145 


10.670 


9.816 


9.071 


12 


23.080 


21.037 


19.282 


16.445 


14.274 


57 


11.290 


10.808 


10.359 


9.550 


8.843 


13 


22.936 


20.922 


19.191 


16.386 


14.234 


58 


10.923 


10.469 


10.046 


9.282 


8.611 


14 


22.787 


20.804 


19.096 


16.324 


14.193 


59 


10.555 


10.129 


9.731 


9.010 


8.375 


15 


22.633 


20.681 


18.998 


16.259 


14.149 


60 


10.188 


9.789 


9.415 


8.735 


8.136 


16 


22.475|20.555 


18.896 


16.192 


14.102 


61 


9.822 


9.448 


9.098 


8.459 


7.893 


17 


22.313 


20.424 


18.790 


16.121 


14.054 


62 


9.457 


9.108 


8.781 


8.182 


7.649 


18 


22.146 


20.290 


18.681 


16.048 


14.003 


63 


9.096 


8.771 


8.464 


7.903 


7.403 


19 


21.974 


20.151 


18.568 


15.971 


13.950 


64 


8.737 


8.434 


8.149 


7.625 


7.156 


20 


21.797 


20.008 


18.450 


15.891 


13.894 


65 


8.382 


8.101 


7.836 


7.349 


6.908 


21 


21.616 


19.860 


18.329 


15.808 


13.836 


66 


8.032 


7.771 


7.526 


7.070 


6.660 


22 


21.430 


19.708 


18.204 


15.722 


13.775 


67 


7.686 


7.443 


7.217 


6.795 


6.413 


23 


21.239 


19.551 


18.075 


15.632 


13.712 


68 


7.347 


7.124 


6.913 


6.521 


6.167 


24 


21.043 


19.389 


17.941 


15.539 


13.645 


69 


7.013 


6.808 


6.613 


6.251 


5.922 


25 


20.842 


19.223 


17.803 


15.442 


13.576 


70 


6.685 


6.497 


6.317 


5.983 


5.678 


26 


20.635 


19.054 


17.660 


15.341 


13.503 


71 


6.364 


6.191 


6.026 


5.718 


5.437 


27 


20.423 


18.875 


17.512 


15.236 


13.427 


72 


6.049 


5.891 


5.740 


5.457 


5.198 


28 


20.205 


18.693 


17.360 


15.127 


13.347 


73 


5.742 


5.597 


5.459 


5.200 


4.962 


29 


19.982 


18.505 


17.202 


15.014 


13.264 


74 


5.441 


5.310 


5.184 


4.947 


4.729 


30 


19.754 


18.314 


17.040 


14.896 


13.177 


75 


5.148 


4.929 


4.915 


4.699 


4.449 


31 


19.519 


18.116 


16.872 


14.774 


13.087 


76 


4.863 


4.755 


4.651 


4.459 


4.273 


32 


19.279 


17.912 


16.698 


14.647 


12.992 


77 


4.585 


4.488 


4.394 


4.216 


4.050 


33 


19.032 


17.703 


16.520 


14.515 


17.893 


78 


4.315 


4.228 


4.143 


3.982 


3.832 


34 


18.780 


17.487 


16.335 


14.378 


12.789 


79 


4.053 


3.975 


3.899 


3.754 


3.618 


35 


18.521 


17.266 


16.144 


14.235 


12.681 


80 


3.799 


3.729 


3.661 


3.531 


3.409 


36 


18.255 


17.038 


15.948 


14.087 


12.568 


81 


3.553 


3.490 


3.429 


3.313 


3.204 


37 


17.983 


16.803 


15.744 


13.933 


12.450 


82 


3.312 


3.257 


3.203 


3.099 


3.002 


38 


17.703 


16.561 


15.534 


13.773 


12.326 


83 


3.077 


3.028 


2.980 


2.889 


2.803 


39 


17.417 
17.123 


16.312 
16.055 


15.317 
15.093 


13.606 
13.433 


12.196 
12.060 


84 
85 


2.846 


2.803 


2.761 


2.681 


2.605 


40~ 


2.617 


2.580 


2.554 


2.474 


2.408 


41 


16.821 


15.792 


14.861 


13.252 


11.918 


86 


2.391 


2.359 


2.328 


2.268 


2.210 


42 


16.512 


15.519 


14.621 


13.064 


11.768 


87 


2.167 


2.140 


2.114 


2.063 


2.013 


43 


16.195 


15.240 


14.374 


12.868 


11.612 


88 


1.946 


1.923 


1.901 


1.858 


1.817 


44 


15.870 


14.953 


14.119 


12.666 


11.448 


89 


1.728 


1.709 


1.691 


1.655 


1.621 


45 


15.540 


14.659 


13.857 


12.456 


11.279 


90 


1.516 


1.501 


1.485 


1.456 


1.428 


46 


15.204 


14.360 


13.590 


12.241 


11.104 


91 


1.309 


1.296 


1.284 


1.261 


1.238 


47 


14.864 


14.056 


13.317 


12.020 


10.923 


92 


1.109 


1.100 


1.090 


1.072 


1.054 


48 


14.519 


13.746 


13.039 


11.794 


10.737 


93 


.921 


.914 


.906 


.892 


.879 


49 


14.171 
13.820 


13.433 
13.116 


12.757 
12.470 


11.563 


10.546 


94 


.748 


.742 


.737 


.726 


.716 


50 


11.326 


10.349 


95 


.592 


.588 


.584 


.576 


.569 


51 


13.465 


12.795 


12.170 


11.085 


10.148 


96 


.468 


.465 


.462 


.456 


.450 


52 


13.107 


12.471 


11.884 


10.840 


9.942 


97 


.371 


.369 


.367 


.363 


.359 


53 


12.747 


12.143 


11.558 


10.590 


9.731 


98 


.243 


.242 


.240 


.238 


.236 


54 


12.385 


11.812 


11.283 


10.336 


9.515 















158 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XLI. 

Life Annuities by American Experience at the Rates 
Stated. 



xAge 3% 31/2% ' 4% 41/2% 5% 



10 


23.343 


21.225 


19.414 


17.855 


16.504 


14.293 


10 


11 


23.225 


21.133 


10.343 


17.800 


16.461 


14.265 


11 


12 


23.103 


21.038 


19.269 


17.741 


16.415 


14.236 


12 


13 


22.977 


20.940 


19.192 


17.681 


16.366 


14.204 


13 


14 


22.846 


20.839 


19.112 


17.617 


16.316 


14.171 


14 


15 


22.712 


20.733 


19.028 


17.551 


16.263 


14.137 


15 


16 


22.573 


20.624 


18.942 


17.482 


16.207 


14.100 


16 


17 


22.430 


20.510 


18.851 


17.409 


16.149 


14.062 


17 


18 


22.282 


20.393 


18.757 


17.334 


16.088 


14.021 


18 


19 


22.130 


20.271 


18.660 


17.255 


16.024 


13.978 


19 


20 


21.971 


20.144 


18.568 


17.173 


15.957 


13.932 


20 


21 


21.808 


20.013 


18.452 


17.086 


15.886 


13.885 


21 


22 


21.640 


19.878 


18.342 


16.997 


15.813 


13.834 


22 


23 


21.467 


19.737 


18.228 


16.903 


15.735 


13.781 


23 


24 


21.289 


19.592 


18.109 


16.805 


15.655 


13.725 


24 


25 


21.104 


19.442 


17.985 


16.704 


15.570 


13.666 


25 


26 


20.914 


19.286 


17.857 


16.597 


15.482 


13.604 


26 


27 


20.718 


19.124 


17.723 


16.486 


15.389 


13.538 


27 


28 


20.516 


18.957 


17.585 


16.370 


15.292 


13.469 


28 


29 


20.308 


18.784 


17.440 


16.250 


15.191 


13.396 


29 


30 


20.093 


18.605 


17.291 


16.124 


15.084 


13.320 


30 


31 


19.872 


18.420 


17.135 


15.993 


14.973 


13.239 


31 


32 


19.643 


18.229 


16.973 


15.856 


14.857 


13.154 


32 


33 


19.408 


18.030 


16.806 


15.713 


14.735 


13.064 


33 


34 


19.166 


17.826 


16.632 


15.565 


14.608 


12.969 


34 


35 


18.917 


17.614 


16.451 


15.410 


14.475 


12.870 


35 


36 


18.661 


17.395 


16.263 


15.249 


14^336 


12.765 


36 


37 


18.397 


17.169 


16.069 


15.081 


14.191 


12.655 


37 


38 


18.125 


16.935 


15.868 


14.907 


14.039 


12.540 


38 


39 


17.847 


16.695 


15.659 


14.725 


13.881 


12.418 


39 


40 


17.560 


16.446 


15.443 


14.537 


13.716 


12.291 


40 


41 


17.266 


16.190 


15.220 


14.341 


13.544 


12.157 


41 


42 


16.963 


15.926 


14.988 


14.138 


13.365 


12.017 


42 


43 


16.653 


15.654 


14.749 


13.927 


13.179 


11.870 


43 


44 


16.335 


15.374 


14.502 


13.709 


12.984 


11.716 


44 


45 


16.009 


15.087 


14.248 


13.483 


12.783 


11.555 


45 


46 


15.676 


14.791 


13.985 


13.248 


12.574 


11.386 


46 


47 


15.335 


14.488 


13.714 


13.006 


12.357 


11.211 


47 


48 


14.987 


14.177 


13.436 


12.757 


12.133 


11.028 


48 


49 


14.632 


13.859 


13.151 


12.500 


11.901 


10.838 


49 


50 


14.271 


13.535 


12.858 


12.236 


11.662 


10.640 


50 


51 


13.905 


13.204 


12.559 


11.965 


11.416 


10.436 


51 


52 


13.533 


12.868 


12.255 


11.688 


11.164 


10.226 


52 



LIFE ANNUITIES — SINGLE LIVES. 



159 



TABLE NO. XLI. 

Life Annuities by American Experience at the Kates 
Stated. 



x Age 



3% 



y 2 % 4 % 



4% % 



5% 



6 % i Age 



53 


13.157 


12.526 


11.944 


11.405 


10.905 


10.009 


53 


54 


12.776 


12.180 


11.628 


11.116 


10.640 


9.786 


54 


55 


12.393 


11.830 


11.307 


10.822 


.10.370 


9.556 


55 


56 


12.006 


11.475 


10.982 


10.523 


10.095 


9.321 


56 


57 


11.617 


11.118 


10.653 


10.219 


9.814 


9.081 


57 


58 


11.226 


10.758 


10.321 


9.912 


9.530 


8.836 


58 


59 


10.835 


10.396 


9.985 


9.601 


9.241 


8.586 


59 


60 


10.443 


10.032 


9.648 


9.288 


8.949 


8.332 


60 


61 


10.051 


9.668 


9.309 


8.972 


8.654 


8.074 


61 


62 


9.660 


9.304 


8.969 


8.654 


8.357 


7.813 
7~549 


62 


63 


9.272 


8.941 


8.630 


8.336 


8.059 


63 


64 


8.885 


8.579 


8.290 


8.017 


7.759 


7.283 


64 


65 


8.502 


8.219 


7.952 


7.699 


7.459 


7.016 


65 


66 


8.123 


7.863 


7.616 


7.381 


7.159 


6.747 


66 


67 


7.749 


7.510 


7.282 


7.066 


6.861 


6.479 


67 


68 


7.381 


7.161 


6.952 


6.754 


6.564 


6.212 


68 


69 


7.020 


6.819 


6.627 


6.445 


6.270 


5.945 


69 


70 


6.666 


6.482 


6.307 


6.140 


5.980 


5.681 


70 


71 


6.319 


6.152 


5.993 


5.840 


5.694 


5.420 


71 


72 


5.981 


5.829 


5.685 


5.546 


5.413 


5.163 


72 


73 


5.651 


5.514 


5.383 


5.257 


5.136 


4.908 


73 


74 


5.328 


5.205 


5.086 


4.972 


4.863 


4.656 


74 


75 


5.011 


4.900 


4.794 


4.691 


4.592 


4.406 


75 


76 


4.699 


4.600 


4.505 


4.413 


4.325 


4.157 


76 


77 


4.392 


4.304 


4.219 


4.138 


4.059 


3.908 


77 


78 


4.088 


4.011 


3.936 


3.864 


3.794 


3.660 


78 


79 


3.790 


3.722 


3.656 


3.593 


3.531 


3.413 


79 


80 


3.496 


3.437 


3.380 


3.324 


3.270 


3.167 


80 


81 


3.209 


3.158 


3.108 


3.060 


3.013 


2.924 


81 


82 


2.928 


2.884 


2.842 


2.801 


2.761 


2.683 


82 


83 


2.652 


2.615 


2.580 


~ 2.545 


2.511 


2.445 


83 


84 


2.379 


2.348 


2.318 


2.289 


2.261 


2.205 


84 


85 


2.107 


2.082 


2.057 


2.033 


2.010 


1.964 


85 


86 


1.839 


1.819 


1.799 


1.780 


1.761 


1.724 


86 


87 


1.579 


1.563 


1.548 


1.532 


1.517 


1.488 


87 


88 


1.334 


1.322 


1.310 


1.298 


1.286 


1.263 


88 


89 


1.103 


1.094 


1.085 


1.076 


1.067 


1.050 


89 


90 


.880 


.874 


.867 


.861 


.854 


.842 


90 


91 


.663 


.658 


.654 


.649 


.645 


.637 


91 


92 


.459 


.457 


.454 


.451 


.449 


.443 


92 


93 


.294 


.292 


.291 


.289 


.288 


.285 


93 


94 


.139 


.138 


.137 


137 


.136 


.135 


94 



160 



FINANCE AND LIFE INSURANCE. 



TABLE NO. XLII. 
Life Annuities by Carlisle Experience at tbe Rates Stated. 



x Age 4 % 5 % 



6 % x Age 4 % 



5% 



6% 






14.282 


12.083 


10.439 


52 


12.258 


11.154 


10.208 


1 


16.555 


13.995 


12.078 


53 


11.945 


10.892 


97988 


2 


17.726 


14.983 


12.925 


54 


11.627 


10.624 


9.761 


3 


18.715 


15.824 


13.652 


55 


11.300 


10.347 


9.524 


4 


19.231 


16.271 


14.042 


56 


10.966 


10.063 


9.280 


5 


19.592 


16.590 


14.325 


57 


10.626 


9.771 


9.027 


6 


19.745 


16.735 


14.460 


58 


10.286 


9.478 


8.772 


7 


19.790 


16.790 


14.518 


59 


9.963 


9.199 


8.529 


8 


18.704 


16.786 


14.526 


60 


9.663 


8.940 


8.304 


9 


19.691 


16.742 


14.500 


61 


9.398 


8.712 


8.108 


10 


19.583 


16.669 


14.448 


62 


9.136 


8.487 


7.913 


1 


19.459 


16.581 


14.384 


63 


8.872 


8.258 


7.714 


2 


19.335 


16.494 


14.321 


64 


8.593 


8.016 


7.502 


3 


19.209 


16.406 


14.257 


65 


8.307 


7.765 


7.281 


4 


19.082 


16.316 


14.191 


66 


8.010 


7.503 


7.049 


5 


18.955 


16.227 


14.126 


67 


7.700 


7.227 


6.803 


6 


18.836 


16.144 


14.067 


68 


7.380 


6.941 


6.546 


7 


18.721 


16.066 


14.012 


69 


7.049 


6.643 


6.277 


8 


18.606 


15.987 


13.956 


70 


6.709 


6.336 


5.998 


9 


18.486 


15.904 


13.897 


71 


6.358 


6.015 


5.704 


20 


18.362 


15.817 


13.835 


72 


6.025 


5.711 


5.424 


1 


18.232 


15.726 


13.769 


73 


5.724 


5.435 


5.170 


2 


18.094 


15.628 


13.697 


74 


5.458 


5.190 


4.944 


3 


17.950 


15.525 


13.621 


75 


5.239 


4.989 


4.760 


4 


17.801 


15.417 


13.541 


76 


5.024 


4.792 


4.579 


5 


17.645 


15.303 


13.456 


77 


4.825 


4.609 


4.410 


6 


17.486 


15.187 


13.368 


78 


4.622 


4.422 


4.238 


7 


17.320 


15.065 


13.275 


79 


4.393 


4.210 


4:040 


8 


17.154 


14.942 


13.182 


80 


4.183 


4.015 


3.858 


9 


16.997 


14.827 


13.096 


81 


3.953 


3.799 


3.656 


30 


16.852 


14.723 


13.020 


82 


3.746 


3.606 


3.474 


1 


16.705 


14.617 


12.942 


83 


3.534 


3.406 


3.286 


2 


16.552 


14.506 


12.860 


84 


3.329 


3.211 


3.102 


3 


16.391 


14.387 


12.771 


85 


3.115 


3.009 


2.909 


4 


16.219 


14,260 


12.675 


86 


2.928 


2.830 


2.739 


5 


16.041 


14.127 


12.573 


87 


2.776 


2.685 


2.599 


6 


15.856 


13.987 


11466 


88 


2.683 


2.597 


2.5-15 


7 


15.666 


13.843 


12.345 


89 


2.577 


2.495 


.2417 


8 


15.471 


13.695 


12.239 


90 


2.416 


2.339 


2.266 


9 


15.272 


13.542 


12.120 


91 


2.398 


2.321 


2.248 


40 


15.074 


13.390 


12.002 


92 


2.492 


2.412 


2.337 


1 


14.883 


13.245 


11.890 


93 


2.600 


2.518 


2.440 


2 


14.695 


13.101 


11.779 


94 


2.650 


2.569 


2.492 


3 


14.505 


12.957 


11.668 


95 


2.674 


2.596 


2.522 


4 


14.309 


12.806 


11.551 


96 


2.628 


2.555 


2.486 


5 


14.105 


12.648 


11.428 


97 


2.492 


2.428 


2.368 


6 


13.889 


12.480 


11.296 


98 


2.332 


2.278 


2.227 


7 


13.662 


12.301 


11.154 


99 


2.087 


2.045 


2.004 


8 


13.419 


12.107 


10.998 


100 


1.653 


1.624 


1.598 


9 


13.153 


11.892 


10.823 


101 


1.210 


1.192 


1.175 


50 


1 12.869 


1 11.660 


1 10.631 


1102 


1 .762 


1 .753 


1 .744 


51 


1 12.566 


1 1.1.410 


| 10.422 


103 


.321 


1 .317 


| .314 



NET SINGLE PREMIUMS. 



161 



TABLE NO. XLIII. 

Single Premiums for an Insurance of 1 at Rates and by the Tables Named. 



M 

> 

M 
S 


American 
Experience 
3 Per Cent 


American 
Experience 
3 1-2 Per 
Cent 


Si I 

111" 


American 
Experience 
4 1-2 Per 
Cent 


Actuaries or 
Combined 
Experience 
3 Per Cent 


Actuaries or 
Combined 
Experience 
3 1-2 Per 
Cent 


Actuaries or 
Combined 
Experience 
4 Per Cent 


H 

> 
2 


10 


.29098 


.24844 


.21485 


.18803 


.29064 


.24747 


.21332 


10 


11 


.29443 


.25154 


.21759 


.19045 


.29456 


.25164 


.21671 


11 


12 


.29798 


.25474 


.22043 


.19295 


.29863 


.25484 


.21993 


12 


13 


.30165 


.25806 


.22339 


.19557 


.30285 


.25866 


.22344 


13 


14 


.30545 


.26150 


.22647 


.19830 


.30718 


.26267 


.22708 


14 


15 


.30937 


.26507 


.22968 


.20115 


.31165 


.26681 


.23086 


15 


16 


.31340 


.26877 


.23301 


.20413 


.31625 


.27108 


.23478 


16 


17 


.31758 


.27260 


.23648 


.20724 


.32099 


.27543 


.23884 


17 


18 


.32189 


.27658 


.24010 


.21050 


.32586 


.28006 j 


.24305 


18 


19 


.32634 


.28070 


.24386 


.21389 


.33086 


.28476 


.24740 


19 


20 


.33094 


.28497 


.24778 


.21745 


.33600 


.28901 


.25191 


20 


21 


.33568 


.28941 


.25185 


.22115 


.34128 


.29460 


.25655 


21 


22 


.34057 


.29399 


.25608 


.22502 


.34669 


.29974 


.26138 


22 


23 


.34562 


.29874 


.26047 


.22905 


.35226 


.30505 


.26636 


23 


24 


.35081 


.30365 


.26504 


.23325 


.35707 


.31050 


.27156 


24 


25 


.35618 


.30874 


.26979 


.23764 


.36383 


.31614 


.27682 


25 


26 


.36172 


.31401 


.27474 


.24223 


.36985 


.32192 


.28231 


26 


27 


.36743 


.31947 


.27989 


.24700 


.37603 


.32789 


.28799 


27 


28 


.37332 


.32512 


.28521 


.25199 


.38236 


.33404 


.29386 


28 


29 


.37939 


.33096 


.29075 


.25719 


.38886 


.34036 


.29992 


29 


30 


.38564 


.33702 


.29651 


.26261 


.39552 


.34687 


.30617 


30 


31 


.39209 


.34328 


.30250 


.26825 


.40235 


.35356 


.31262 


31 


32 


.39873 


.34976 


.30871 


.27416 


.40936 


.36045 


.31929 


32 


33 


.40558 


.35647 


.31517 


.28030 


.41654 


.36754 


.32617 


33 


34 


.41263 


.36339 


.32186 


.28669 


.42389 


.37482 


.33327 


34 


35 


.41988 


.37055 


.32881 


.29335 


.43144 


.38235 


.34060 


35 


36 


.42736 


.37795 


.33602 


.30029 


.43917 


.38996 


.34817 


36 


37 


.43504 


.38560 


.34350 


.30751 


.44710 


.39817 


.35599 


37 


38 


.44295 


.39259 


.35125 


.31503 


.45524 


.40617 


.36407 


38 


39 


.45107 


.40164 


.35927 


.32283 


.46358 


.41458 


.37241 


"39 


40 


.45942 


.41004 


.36757 


.33095 


.47214 


.42323 


.38104 


40 


41 


.46800 


.41869 


.37617 


.33937 


.48093 


.43217 


.38996 


41 


42 


.47680 


.42762 


.38566 


.34812 


.48995 


.44137 


.39919 


42 


43 


.48583 


.43681 


.39425 


.35719 


.49919 


.45083 


.40871 


43 


44 


.49510 


.44628 


.40375 


.36660 


.50863 


.46055 


.41852 


44 


45 


.50458 


.45601 


.41355 


.37635 


.51826 


.47048 


.42857 


45 


46 


.51430 


.46600 


.42366 


.38643 


.52804 


.48060 


.43886 


46 


47 


.52423 


.47626 


.43406 


.39685 


.53795 


.49088 


.44935 


47 


48 


.53437 


.48677 


.44476 


.40759 


.54798 


.50130 


.46002 


48 


49 


.54470 


.49752 


.45574 


.41867 


.55812 


.51190 


.47088 


49 


50 


.55522 


.50849 


.46. '599 


.43004 


.56836 


.52264 


.48191 


50 


51 


.56589 


.51967 


.47848 


.44169 


.57870 


.53347 


.49311 


51 


52 


.57671 


.53104 


.49021 


.45363 


.58912 


.54447 


.50446 


52 


53 


.58767 


.54258 


.50215 


.46582 


,59960 


.55555 


.51595 


53 


54 


.59874 


.55430 


.51431 


.47826 


.61015 


.56670 


.52757 


54 


55 


.60992 


.56615 


.52665 


.49093 


.62075 


.57797 


.53931 


55 


56 


.62118 


.57813 


.53915 


.50380 


.63139 


.58931 


.55116 


56 


57 


.63251 


.59022 


.55181 


.51687 


.64205 


.60070 


.56310 


57 


58 


.64389 


.60239 


.56459 


.53010 


.65274 


.61215 


.57514 


58 


59 


.65530 


.61464 


.57748 


.54348 


.66344 


.62350 


.58726 


59 


60 


.66672 


.62693 


.59046 


.55699 


I .67414 


.63517 


.59942 


60 


61 


.67813 


.63924 


.60349 


.57059 


! .68480 


.64668 


.61163 


61 


62 


.68950 


.65155 


.61656 


1 .58426 


| .69541 


.65816 


.62383 


62 


63 


.70083 


.66383 


.62963 


.59797 


1 .70595 


.66959 


.63600 


63 


64 


.71208 


1 .67607 


.64269 


.61170 


.71640 


.68096 


.64812 


64 


65 


.72324 


1 .68824 


.65570 


.62541 


.72673 


.69224 


.66017 


65 


66 


.73427 


| .70030 


.66863 


.63908 


.73694 


.70338 


.67212 


66 


67 


.74516 


' .71223 


.68145 


.65265 


.74700 


.71440 


.68396 


67 


68 


.75588 


1 .72401 


.69414 


.66611 


! .75689 


.72526 


.69565 


68 


69 


.76641 


.73560 


.70665 


.67943 


| .76662 


.73766 


.70719 


69 


70 


.77673 


1 .74698 


.71896 


.69254 


| .77617 


.74650 


I .71857 


70 



CHAPTER VIII 
Life Contingencies and Money 

If a certain sum, say $1000.00, be promised and be made payable 
at a certain time, say in twenty years, its present value at a given rate 
per cent, say 4%, would be $456.39. This result is reached by the 
process of discounting illustrated in the Chapter on Interest (Chapter 
V) Article 25 et seq. If the same sum be promised and be made 
payable upon the happening of the death of A, now aged thirty-five 
years, its present value computed on the American experience Table 
of Mortality at the rate of four per cent would be $328.81. This 
benefit is called a life insurance. The result last stated is arrived at 
by a process of discounting, but one of the factors, that of time, differs 
from the case first instanced in that instead of employing a time 
certain during which the rate of interest is applied, we must use also 
the probability of living, or rather of dying, of A. This probability 
is determined by means of a mortality table, of which a number of 
examples are given in this book. A process which may be used in 
finding the present value of the $1000.00 insurance promised is as 
follows : 

The group of lives treated in the American Experience table of 
mortality numbers 100000, called the radix of the table, beginning at 
age ten. At the age given, 35, there remain living of this 100000 
group, 81822, of whom 732 will die during the current year. Hence, 
$732000 must be paid their beneficiaries at the end of the year. It 
follows that for this first year 81822 persons must pay in advance a 
sum which improved at 4% interest, will amount to $732000 at the 
end of the year. Discounting $732000 at 4% one year, we have 
$703846.24, which, divided by 81822, gives $8,602, the sum to be 
contributed by each person for the first year's insurance. Between 
the ages 36 and 37, there will die of the group, 737, making it necessary 
to pay beneficiaries $737000 at the end of that year. This sum dis- 
counted two years gives $681393.40 or $8,327 per member which also 
must be paid down at the beginning. For the third year, $742000 
must be provided, making it necessary to pay down $659560.80 or 
$8,061 per member and so on to the end of the table. The total of 
these discounted sums has been computed and is $328,809 for each 
$1000 of insurance. This latter sum is the present value and also the 
net cost of $1000 life insurance at age 35 according to the rate of 
mortality shown by the American Experience table and 4% compound 
interest. It is hence called a Single Premium. This premium is 
usually represented by the symbol Ax. Life Annuities are closely 
related to life insurances and their values may be computed by a very 
similar process. We may take the same group and age. At the 
end of the first year there will survive of the original 81822 only 81090, 
732 having died as already stated. If the annuity be one dollar, 
then there must have been paid down at the beginning of the year an 



COMPUTING ANNUITIES AND INSURANCES. 163 

amount which at 4% interest would pay 81090 dollars at the end of 
the year. This would be the last mentioned sum discounted one 
year at 4% and is $77971 or $.9524 for each of the members of the 
group. The number living at the end of the second year and to each 
of whom a dollar must be paid, is 80353, requiring $80353. To provide 
for this, the latter sum discounted two years should have also been 
paid down at the beginning. This process is continued to the end of 
the table by which means it is found that the total amount required 
to be paid at the beginning by each member of the original group is 
$16,451. This then is the present value and also the net cost of a 
life annuity of one dollar at age 35 by the table and rate of interest 
used in the computation. The value is usually represented by the 
6ymbol ax. If instead of proceeding as above, we had divided the 
number living at the end of the year by the number of the original 
group at the beginning of the year, we should have had the probability 
of living one year at age 35, which is .991054. Also if we divide the 
number dying during the year, 732, by the original group 81822, we 
will have the probability of dying during the year, which is .008946. 
The sum of these probabilities it will be observed, is unity or certainty, 
since man must either live or die and the two factors cover all of the 
contingencies. The probabilities for the succeeding years may be 
found in the same way. These probabilities have been tabulated and 
are given in connection with the mortality tables published in this 
book. If instead of dealing with the whole group of lives, we had 
taken a single life, we might have found the values of the insurance 
and annuity as well. For instance, the probability of a given person 
dying between the ages 35 and 36 is, .008946. As the insured must 
pay for his insurance, he, on a $1000.00 policy, should pay at the 
beginning of the year the value of the chance or probability that the 
company will have to pay his beneficiary $1000. That is, $1000 
multiplied by the probability of its becoming due, .008946, which 
gives $8,946. This sum must be discounted or divided by $1.04, 
which gives $8,602, as the sum which must have been paid at the 
beginning of the year, as before. The contribution to be made for the 
subsequent years may be found in the same way, having due regard 
to the probability of dying and the number of years discount to be 
used. In the case of annuities, we must use the probability of living 
instead of that of dying, because the value of each of the one dollar 
payments depends not only on the number of years which will intervene 
before it is to be received, but also upon the probabilit;/ of the nominee 
living to receive it. 

While the foregoing discussion illustrates the principles underlying 
life insurance and life annuities, it is apparent that the processes em- 
ployed are too cumbersome for practical purposes. They may be em- 
ployed, however, in deriving other functions which may be used with 
great facility in computing many of the benefits depending upon life 
contingencies. 



164 FINANCE AND LIFE INSURANCE. 

If the process suggested for finding the present value of the life 
insurance be represented in the form of an equation, using 1 for the 
number living, d for the number dying, x for the age of the insured, 
px for the probability of living and qx for the probability of dying at 
a given age, and n for a number of years, it would appear as follows: 
The net single premium, A35 or 

$732,000 $737000 $742000 

$328 . 809 = 1 1 and so on 

81822X1.04 81822 X(1.04) 2 81822 X (1 .04) 3 

to the end of the table. Each of the numbers on the right hand might 
have been written in this form, viz: 

732 1 

X X ($1000), performing the indicated operation the factors 

81822 1.04 

become respectively . 008946 X .961538X1000 which read in the light 
of the notation above suggested, will be 

dxXvx 

X1000, or if the policy be assumed to be 1, the equation finally 

lx 

becomes, Ax = . 3288 = 

V*dx v* +1 dx + l v* +2 dx+2 

1 1 (- years to ends of the table, 

lx lx lx 

This series will be adequate to compute the single premium of a policy 
of 1, for age 35, but it would not be available for computing the premium 
at any other age, for instance, at age 34 or 36, 

A formula may be derived, however, which is general in its applica- 
tion, as follows: 

If we begin with the first year of the table, it will appear in the form 

vdx +v 2 dx + 1 +v 3 dx +2 +v 4 dx +3 

Ax = h & etc, 

lx 

By multiplying both the numerator and the denominator of the 
right hand member of said equation by v, it will appear as follows: 
v* +1 dx+v* +2 dx + l+v* +3 dx+2 

Ax = & etc. 

v x lx 

The denominator of the above equation has been called D, adding the 
subscript x to denote the age. In the above case, the x represents 10 
that being the youngest age of the table and the number living is then 
100000. The quantities in the numerator v x+1 dx, +v x+2 dx + l have 
been called Cx, Cx + 1, Cx+2 and so on and the equation at age 10 
would be written: 



CONSTRUCTION OF COMMUTATION COLUMNS. 165 

Cx+(Cx+l) + (Cx+2) 

Ax= h & etc. 

Dx 

(Cx+l)+(Cx+2)+(Cx+3) etc. 

At age 11, it would be Ax-fl = 

Dx + 1 

and so on for older ages throughout the table. 

Since all the years of the life after ten are contained in the first 
equation it is apparent that if the quantities C be summed from the 
oldest age down to ten and the numbers recorded at each addition 
we vail have the numerators of all the equations which we would have 
had we extended those above given to all the years to 95, the end 
of the table. The numerator columns thus formed have been called 
M columns by the actuaries and they are so marked in the tables 
here published. The D column has been formed for each year by 
computing and recording the quantities v x lx, v x+1 lx + l, v x + 2 lx+2 
and so on to the end of the table. Now, to solve the first equation 
above, we would simply divide M, the equivalent of v x+1 dx+v x+2 
dx + 1 +v x+3 dx+2 and etc. by Dx which is the equivalent of v x lx. 
The quotient will be Ax or $213.99. For age eleven, Mx + 1 divided 
by Dx + 1 would give the net single premium for that age and so on 
for all the ages given in the table from which we may derive the general 

Mx. 
formula, Ax = — ■ 
Dx 
By a similar process, a table Nx is derived for findng the value of 
annuities but instead of using dx, the number dying, as a factor, the 
number living at the end of each year is employed. That is, beginning 
at the first year, lx + 1, lx+2 and so on, are substituted in the original 
equation. Thus, the formula for ox, a life annuity, is derived and is 
as follows : Nx 

ax = 

Dx 

The tables which have just been described are called "Commutation" 
Tables. In some cases, two other columns marked respectively S and 
R have been computed. These may be explained as follows: If at a 
given age an annuity of 1 were purchased to be increased by 1 each 
year, to find its present value, we should have the following equation : 

An increasing annuity or Nx Nx + 1 Nx+2 

(lax) = 1 1 (-and etc. 

Dx Dx Dx 

For convenience, the quantities Nx have been summed and are called 
S, and therefore we may use the formula for an annuity of 1, increasing 
by 1 each year, Sx 

lax = 

Dx 



166 FINANCE AND LIFE INSURANCE. 

For increasing insurances, the M columns have been summed in 
the same manner forming the new column R. Examples of these 
columns will be found in the insurance commutation columns. Other 
tables have been computed and published here for further reducing 
the labor of computing values based on life contingencies, and their 
uses will presently be explained, and illustrated, In books such as 
Principles and Practice of Life Insurance, Various Derived Tables 
and other works, many of the values have been computed, tabulated 
and published for the convenience of the insurance profession. We 
will now state, without demonstration, a number of formulas and rules 
for solving problems involving life contingencies and illustrate them 
by examples. 



CHAPTER IX 
Of Life Annuities 

1. The present value of a life annuity of 1 is found by dividing 
the quantity N of the given age by the quantity D at the same age, 

Nx (1) 

thus, ax = . 

Dx 

2. What is the present value of an annuity of $250.00 at age 35, 
computing on the American Experience table of Mortality, at 4% 
interest? 

Solution: By the commutation table (No. 37) in the N column 
opposite age 35, we find the figures 341109.86 and D35 = 20734 . 96, 
and by formula (1) 341109.86 

a35 = = 16 . 451 which, multiplied by 250 

20734.96 
gives 4112.75. Answer, $4112.75. 

3. A Temporary Annuity is found by dividing the difference 
between the N functions of the age at the beginning and the end of 
the given term by the D function of the given age, thus: 

N'x-Nx+n 

|nax = (2) 

Dx 

4. Example: What is the present value of an annuity of $250 
for the term of ten years at age 35, computed upon American Experience 
table at 4%? 

Solution: From the 4% commutation table, we obtain the fol- 
lowing values : N35 = 341 109 . 86 ; N45 = 180921 . 20, D35 = 20734 . 96, 
and by formula (2) 341 109 . 86 - 180921 . 20 

|10c35= =7.7255 

20734.96 

$250X7.725 =$1931.25, Answer. 

5. A Deferred Annuity is found by dividing the function N at 
the age when the annuity is to be entered upon by the function D at 
the age when the annuity is contracted. Thus: 

Nx+n 

n|ax=- (3) 

Dx 

6. What is the present value of a life annuity of $250, deferred 
ten years at age 35, computed upon American Experience table at 
4%? 



168 FINANCE AND LIFE INSURANCE, 

Solution: N 45 = 180921 .20, D 35 =20734.96, and by formula 
(3), 180921.20 

10 1 a 35 = =8.7254 which, multiplied by 250 gives 

20734.96 

2181.35. Answer, $2181.35. 

7. Note — By adding the temporary and deferred annuities above, 
we find their sum to be the same as the whole life annuity, thus 
7 . 7255 +8 . 7254 = 16 . 451. It thus appears that the term and deferred 
annuities supplement each other and that the following equations 
hold: 

ax — |n#x=n| ax. 

ax— n\ax = \nax and \nax+n\ax = ax. 

8. A Deferred Temporary Annuity is found by dividing the 
difference between the N functions for the age at the beginning and 
the end of the deferred term by the function D for the given age, thus: 

Nx+n— Nx+m+n 

n|ma x = (4) 

Dx 

9. Example: What is the value of annuity of $250 for term of 
fifteen years entered upon ten years from the date of the contract, 
the annuitant being 35 years of age, computed on the American Ex- 
perience table at 4%? 

Solution: N 45 = 180921 . 20, N 60 =53118.75 and D 35 = 20734 . 96, 
and by the formula, 

180921.20-53118.75 

10|l5o 35 = X250 = (6. 163X250) =1540.75, 

20734.96 
Answer, $1540.75 

10. Ordinary life annuities are payable at the end of each year 
survived by the life on which it is based. Such are the annuities 
already considered. But the contract or law governing may provide 
for payment in the same proportion for the fractional part of the last 
year of the life. Such an annuity is called a Complete Annuity 
and is sometimes represented by the symbol ax- 
il. On the assumption that the death rate is uniform throughout 

the year, the value of a complete annuity is that of an ordinary annuity 
plus an insurance equal to one-half the annual payment, payable at 
the moment of death. The value of an insurance has already been 
assigned the symbol A. But an insurance so marked is payable at 
the end of the year of the death which, it is assumed in insurance 
calculations, will occur on the average at the middle of the year, so 
that the half single premium must be increased so that it equals the 
first payment of a sum payable semi-annually at the nominal rate 



COMPLETE AND FRACTIONAL ANNUITIES. 109 

This would be represented by the equation 
A(l+i)» 

a x = Ox H . (5) 

2 

12. As an example, let it be required to find the value of a complete 
annuity at age 35, using American Experience table and 3 1 / 2 %. We 
have; q 3 5 = 1 7.614. A 35 = .3706. And, 

Va X .3706 X V 1 .035 = l /s X .3706 Xl .017344 = . 377026-=- 2 = . 1885. 
And aso = 17 . 614 + . 1885 = 17 . 8125. 

13. In the last example, it will be noticed that the amount of 1 at 
the actual rate for the half year, 1.0173, does not differ much from 
the amount of 1 at the nominal rate for one-half year, 1.0175, and it 
is considered sufficiently accurate in practice to multiply by 1 plus 
one-half the nominal rate for yearly annuities. 

14. The formula for a complete half-yearly annuity is sometimes 
written, a ° ( l } =a f i ) + 1 / 4 A x (l- r -i) ( * ) . This formula will be better un- 
derstood after reading the next section. 

15. An annuity may be made payable in 2, 4 or m installments 
and the calculation of the exact value of such annuities presents an 
interesting problem, but one of little practical value here. A useful 
approximate formula is stated as follows: m — 1 

o C ?=Ox+ — (6) 

2m 

In this expression, m represents the number of installments into 
which the year's payment is divided. Thus, if the annuity is 1 payable 
in semi-annual payments, that is in 2 payments. The formula becomes 



/2-l\ 

*x (a) =aJ 1=74 

\2X2/ 



ax -HA- For 4 installments, we have 

4-1 

a 1 * = o x H = 3 /s = a x + 3 /jj. For monthly installments, we have 

2X4 

-1 1 

ox + l = — l=a x -\ and so on. 



/12-1 11\ 11 

( = -)=**+- 

\2X12 24/ 24 



16. The value of an annuity of 1 at age 35, payable in half yearly 
installments would by the above formula, be 17.614 + .25 = 17.864; 
and payable quarterly, the value would be 17.614+ .375 =17.989. 

17. When m in the above formula is supposed to become infinite 
the correction m — 1 

approaches 1 / 2 as its limit and the value of the 

2m 



170 FINANCE AND LIFE INSURANCE. 

annuity becomes a K + l / 2 . Such annuities are called Continuous 
Annuities, by which it is meant that the installments are payable 
momently. The reader will have little occasion for the use of the latter 
formula. The symbol assigned by actuaries to the continuous annuity 
is ax. 

18. An annuity on two or more lives is called a joint life annuity. 
Such an annuity terminates with the first death. It is possible to 
compute commutation tables for such annuities but the numerous 
combinations of ages which have to be considered are such that so 
far as the writer knows it has not been undertaken for more than two 
lives. Annuities on two joint lives have been computed, but only for 
ages taken at intervals and not year by year. Tables on the North- 
ampton, Actuaries and some others have been so treated. 

19. Single annuities on joint lives may be computed directly 
from the mortality table by means of formulas for approximate sum- 
mation. These will be noticed further on. 

20. The calculation of annuities on joint lives has been greatly 
facilitated by regraduating the fundamental table by what is known 
as Makeham's law. Not all tables of mortality are susceptible of 
regraduation by this process, but some of the most important ones have 
been so treated. The English Twenty Office healthy males table 
was so treated by George King in 1880. Mr. Arthur Hunter, a New 
York actuary and author, regraduated the American Experience 
table in 1902. The Carlisle table was also regraduated in 1880 by Mr. 
King. Mr. Abb Landis, an author and actuary of Nashville, Tenn., 
has "Makehamized" the National Fraternal Congress table. The 
Actuaries or Combined Experience table was also regraduated by 
Makeham's formula in 1915 by the author of this work. 

The effect of this process is to convert the table into a series with 
decrements conforming to certain constants and the force of mortality 
derived from the original table. This latter function may be used to 
find two or more equal ages having a force of mortality equivalent to 
that of two or more lives of different ages, so that it is only necessary 
to compute commutation columns, annuities and premiums for equal 
ages. This method saves an enormous amount of labor and space 
and while annuity and insurance values derived from such tables do not 
exactly represent the values which the original data would produce 
they are sufficiently exact for all practical purposes and the labor 
saved warrants their employment where joint lives are involved. 

We have computed and published here D and N commutation 
tables and annuities at the rates of five and six per cent on two and 
three lives from the regraduated Carlisle table ; and at the rate of three 
and one-half, five and six per cent on two and three fives from the 
regraduated American Experience table. Also at the rates of four, 
five and six per cent on two and three lives from the regraduated 
Actuaries table. One or the other, sometimes two, of these tables 
in their original form, have been adopted as standard tables in nearly 



JOINT LIFE FORMULAS. 171 

all the states of the Union. They, with the single life tables may 
therefore be employed in computing all sorts of values depending on 
one, two or three lives. And by means of the formulas for differencing 
approximate values for four or even five lives may be derived. 

21. Where joint life commutation columns are available, whole 
life, temporary, deferred and deferred temporary annuities are com- 
puted by the same forms of formulas as single life annuities, thus: 

N x , Nx+nJy+n Nxy-Nx+n; y+n 
c xy = ;n|a xy = ; [na xy = and 

-L'xy L'xy -L'xy 

Nx+n:y +n — Nx +n-f-m:y -fn-f-m 

n|m oxy = 

Dxy 

22. A Revisionary Annuity, is one which begins at the death 
of a person now aged y and continues during the life of a person now 
aged x. Its value is the difference between the value of a whole 
life annuity on x and the value of a joint life annuity on x and y 
The formula is a y |x=a x — a xy . (7) 

24. An annuity on three lives and to the survivors successively 
is represented by the symbol a^ and its value is 

2i+fly+fl«- Oxy — a xz — fl yz + O xyz . (8) 

25. An annuity on the survivor of x an y but not to begin until 
the first death is equal to the sum of a reversionary annuity on the 
fife of each, that is, (a x — a x y ) + ( a y — a x y ) = 

Ox+Oy— 2o xy . (9) 

26. An annuity on the life of x provided he shall survive both 
y and z, that is, to begin at the death of both, is, 

o x — «xy — Oxz + a xyz . (10) 

27. The value of an annuity to be shared by x and y during 
their joint lives and by the survivor of them during his life is equal 
to the sum of the values of two annuities on their separate lives less 
the value of a joint annuity on their lives, thus: 

Gxy = a x -\-a y — o xy . (11) 

28. If the payments are postponed until the first death the value 
of the annuity of the last article would become : 

fli + Cy — 2a X y. 

29. The value of the interest of x or y in the annuity of Article 
27 is that of an annuity on his life less one half the value of the joint 
life annuity, thus, for x it is: a x — 1 /2(i xy . (12) 

30. The value of an annuity payable during the joint fives of y 
and z after the death of x, is the value of the joint fife annuity on y 
and z less a joint fife annuity on the three lives, thus: 

a x y8 = a yt — o X yz. (13) 



172 FINANCE AND LIFE INSURANCE. 

31. The value of an annuity on the life of x to begin after the death 
of either y or z is the value of an annuity for the life of x less the value 
of a joint life annuity on the lives of x, y and z, thus: 

0y»|* = Ox — a xyz . (14) 

32. The value of an annuity on the life of x to begin after the 
death of bothy and z is the value of an annuity for the life of x less an 
annuity which is to fail either on the death of x or of both y and z, thus 
Oy^|x = Cx — a x :^i (15) 

33. The value of an annuity on the life of x after an annuity for 
a term certain on the lif e of y is the value of n | ox less a deferred joint 
life annuity on x and y, thus: 

n|a y | r =na| x — n|a xy . , (16) 

34. The value of a joint life annuity on two of three lives, to begin 
only after the first death of the three will be understood by the 
formula: a xy -r-axz+»y« — 3ax yz . (13) 

35. A little study of the principles involved in the foregoing 
formulas will enable the reader to form other combinations necessary 
to solve the problems which will arise in practice. Nearly all of 
those given are of great practical value to those who have occasion 
to value properties and interests involving life contingencies. A 
number of examples will be given illustrating the use of formulas and 
principles on which they are based, in the succeeding articles. 

36. Before employing the foregoing formulas in the solution of 
problems, it will be helpful to explain a method of deriving joint life 
annuities by the Carlisle, American Experience and Actuaries re- 
graduated tables. As stated, these tables have been graduated by 
Makeham's law of mortality and they constitute a series in which the 
force of mortality is the only variable, the other functions involved 
in the process being constants derived from the original data. The 
force of mortality for each year of life is given in tables (Table 27). 
These regraduated tables have the property that enables us to take 
the average force for two, three or more different ages and use it to 
find an equal age at which two, three or more lives will have the same 
value. Thus, (120+^30+^40 = [i33+(i33+[i33, nearly for the force 
of mortality by the table, American Experience, for age 20 is .00786, 
for age 30, .00835 and for age 40, .00977. Their sum is .02598, which 
sum divided by 3 gives .00866, corresponding nearly to age 33 at 
which (i is .00863. By interpolating in a manner similar to that, 
employed with logarithms a closer approximate age may be found. 
Thus, .00866 falls between ages 33 and 34 which have [i = . 00863 
and .00875 respectively and their difference is .00012. The difference 
between .00866 and .00863 is .00003, which is one-fourth of that 
of a whole year which as we have seen is .00012, hence the approxi- 
mate equal age is 33 * / 4 years. 

37. Now let it be required to find the value of an annuity on the 
joint lives of A, B and C, aged respectively, 20, 30 and 40 years. We 



METHODS FOR JOINT LIFE ANNUITIES. 173 

have found the equivalent equal age from the American Experience 
table to be 33V4- We will therefore use annuities derived from that 
table. Using the six per cent table (Table No. XXVIII) we find 
033:33:33 = 10.12045 and 034:34:34 = 10.01075. The difference in the 
annuity for a difference of a whole year is 10 . 12045 — 10 . 01075 = . 10970. 
For one-fourth year, it is . 10970-^4 = .02742. Now since the annuity 
decreases as the age increases, this is to be deducted from the younger 
age and we have 10. 12045- .02742 = 10.09303, which is the required 
annuity. 

38. Let us now compute the values by the Carlisle table, omitting 
the explanatory matter so as to show up the figures more clearly. 



pi20 = . 00674 

\i.30 = . 00942 

^.40 = 01137 

3). 02753 



Average =.00918 
ix27 = .00910 

ti.28 = . 00920 



.00010 



027:27:27 =10.23159 

028: 2 s: 2 8 =10.14654 

Difference for one 



year = .08505 



Difference for 8 /i year = .068040 which 
subtracted from 10.231590 = 



027.8:27.5:27.8 =10.16355 



Hence, 020 :3o :4o = 10 . 16355 



p. 00918= Age 27.8 years. 

39. Let it be required to compute the present value of a term 
annuity on the same lives, using American Experience at 6%. Let 
the time be ten years. The average equal age we have already found 
to be 33 years. The formula is 

N — n N 



From the commutation table (Table No. XX) for three lives 6%, we 
have N 33 :3 3 :33 =85499482, N i3 : i3 : i3 =31125106, and 033:33:33 =8448181. 
Putting these values in the formula and solving we have (85499482 — 
31125106)^8448181=6.4362, the required annuity. 

For a deferred annuity, same ages, we have, 

^3:43:43 



31125106-^8448181=3.6842, which is the value of the de- 



ferred annuity. The sum of these should equal the whole life annuity 
at age 33, and so it does, 10. 1204. 

We might have found the annuities for 33 l U years, by first finding 
them for the equal ages 33 and 34 separately and interpolating as 
illustrated in the two preceding articles. 



174 FINANCE AND LIFE INSURANCE. 

40. The value of a joint life annuity on the lives of A and B is 
found as follows: [jl20 =. 00674, ^30 = .00942. Their sum is .01616 
and their average equal age is [i. .00808 age =25.8 years. 

41. The value of A and C, we find in the same way to be 11 . 18926, 
it being the equal age 34.5:34.5, and the value of B and C, having 
the equal age 36V 4 :36V4 is 10.98245. 

42. From the table of single life annuities, American Experience 
6% (Table XLI), we have values as follows: A, or o 2 o = 13.932; 
B or o 30 = 13 . 320 and C or a 40 = 12 . 291. 

We may use the values now found in solving a variety of practical 
problems and for convenience of reference, they may be collected 
as follows: 

o 2 o = 13.932 o xy = 11.973 o xyz = 10.093 

030 = 13.320 ox, = 11. 189 |l0oxy, = 6.436 

040 = 12.291 o yz = 10.982 10|o xyz = 3.684 

43 What is the present value of the dower interest of a widow 
aged 30 in an estate valued at $100000 producng a net annual income 
of 6% on the value? 

Solution: The dower is one-third of the estate during the life 
of the widow. It would produce one-third of the annual income or 
$2000 per year. This may be valued as an annuity. The present 
value of an annuity of 1 at age 30 is 13.320 and the present value 
of an annuity of 2000 is, of course, 13.320X2000=26640. Answer, 
$26640. 

44. What is the present value of the remainder of the estate 
mentioned in Article 43? 

Solution: The present value of the whole estate, by the problem, 
is $100000. The present value of the remainder is therefore 100000 — 
26640=73360. Answer, $73360. 

45. What is the present value of the inchoate dower right of a 
woman in an estate the value of which is 100000 owned by her husband, 
aged 40, her age being 30, the income of the estate being 6%? 

Solution: The interest or right of the wife is a reversionary 
annuity of $2000, dependent upon her surviving her husband and is 
the same as an annuity on her life less the value of a joint life annuity 
on the lives of herself and husband. A life annuity of 1 at age 30, is 
13.320. A joint annuity of 1 on the lives of two persons aged 30 and 
40 is 10.982. Their difference, 13.320-10.982, is 2.338, which, 
multiplied by 2000=4676. Answer, $4676.00. 

46. What is the present value of the curtesy interest of the hus- 
band of the woman of the preceding problem assuming that he has 
such interest, her real estate being worth $100000, earning an income 
of 6% on its value? 

Solution: Here, again we have to find the value of a reversionary 



PARTITION OF JOINT LIFE ANNUITIES. 175 

annuity. The annuity on his life is 12.291. The joint life annuity 
is 10.982. The difference is 1.304, which, multiplied by 6000 gives 
7824. Answer, $7824.00. 

Note: The value of the courtesy initiate is not considered in this 
problem. If considered, his interest, of course, would be a life annuity. 

47. What is the present value of the interest of A and B in an 
annuity of $5000 bequeathed to them to be enjoyed by them equally 
during their joint lives and by the survivor during his life? 

Solution: The whole value is the same as if each of the nominees 
was to receive an annuity for his life less the value of the joint life 
annuity on their lives; but here we are to find the value of the interest 
of each. First A, whose age is 20, receives an annuity for life less 
one-half the joint annuity which goes to B, that is, 

11.973 

13.932 =7.945. And 7.945X5000 =$39725. B's interest is 

2 

11.973 

13.320 =7.33250, and the latter sum multiplied by 5000 = 

2 

36662.50 We have, therefore, A's share =$39727.50. B's share = 
$36662 . 50 and the whole value a 2 o + oso -020 '.30 =$76390.00, the sum of 
the interests of A and B. 

48. What would be the present value of the interests of A and B 
in the annuity of the preceding problem on the supposition that the 
payments are not to begin until the death of one of the nominees. 

Solution: The values of the separate interests of A and B are 
found by the method illustrated in Article 46. A's interest being 
(13.932-11.973) X5000 $9795.00 and B's being (13.320-11 .973) X 
5000, $6735.00. The whole being (a x + c y -2 0xy ) X 5000 =$16530.00. 

49. Let it be required to find the present value of an annuity of 
$1500.00 bequeathed to A and B jointly subject to a life interest in 
C, that is, to be entered upon at the death of C. 

Solution: So long as C lives he receives the annuity and A and 
B get nothing. Also when either A or B dies, the annuity expires. 
Hence, the value of the interest of A and B is a xy -a xyI . That is 
(11 . 973 - 10 . 093) X 1500 = 2820. Answer, $2820 . 00. 

50. Let the preceding problem be so modified that after the death 
of C the annuity goes to A and B and to the survivor of them, what 
then is the present value of their shares of the whole? 

Solution: First to find the value of A's share, it is obvious that 
he has the value of an annuity on his life less the value of the interest 
of C, the first taker, and of that of his co-nominee, B. This is repre- 
sented by Gx-0 X2 -V 2 0x y + 1 /20xyz. 



176 FINANCE AND LIFE INSURANCE. 

From the figures derived above, we have, 
(13. 932-11. 189- 1 / 2 (ll-973)+ 1 /2(10. 093) ) X 1500 =2704.50. B's 

share is 13.320 -10.982 -7 2 (11.973) +72(10.093) X1500 =2097.00. 
The whole value is 2704 . 50 +2097 . 00 = 4801 . 50. Answer, $2704 . 50 
$2097.00 and 84801.50. 

51. Let the problem be so modified that A and B enjoy the annuity 
until the death of both, when it shall go to C. What then would be 
the present value of the interest of C? 

Solution: So long as A or B live, C gets nothing. His annuity 
is therefore one on his life less the value of the annuity during that 
part of his life passed in connection with the lives of A and B and the 
survivor of them. That is, a z — a z ixy, which is equivalent to a 2 — 
Oxz + ayz — « X yz for which we have the values (12.291—11.189 + 
10 . 982 - 10 . 093) X 1500 = 2986 . 50. Answer, $2986 . 50. 

52. Again let us now suppose that the annuity goes to all and to 
the survivors and to the last survivor, what is its present value? 

Solution: This value is expressed 'by the formula a^ = a x + a y + 
Oz — flxy- fl xz — Oya + Oxyz. For these, we have derived the value s 
13.932 + 13.320 + 12.291-11.973-11.189-10.982 + 10.093=15.492 
which, multiplied by 1500, gives 23238. Answer, $23238.00. 

53. What would be the present values of the several shares of the 
three nominees in the annuity of the last problem? 

Solution: First take A's interest. During the period preceding 
the first death he receives one-third of the annual payments, which 
is a xy zX500. If he be one of the survivors, he will then receive 
<2xyX750 or a xz X750. The value of which is 750(a z | X y + Oy|xz). 
If he again survives, he will receive the whole annuity during his life. 
The value of this contingent benefit we have seen is a yz | x . These 
several values may be reduced to the form a x — 72(«xy+«xz) +73«x yz , 

the values 10.093 

of which are 13.932-7 2 (11.973+11 . 189) + X 1500 =8573. 

3 

In the same way we find B's share to be 13 . 320 + 7 2 (H • 973 + 10 . 982) + 

10.093 10.093 

X 1500 = 7810. 25 and C's, 12.291 - 1 / 2 (11. 189+10.982+ 

3 3 

X 1500 =6854 . 75. The sum of these values is 23238, as by Article 52, 
we have found it should be. 

54 : Again, let us suppose the annuity of the preceding sections to 
be the income of an estate valued at $25000 which is enjoyed by all and 
the survivors, the whole estate to go to the last survivor. What is 
the present value of the annuity and of the remainder? 

Solution: The present value of the whole of course, is $25000. 
We may find the required values by regarding the estate as an in 



PROBLEMS INVOLVING THREE LIVES. 177 

surance payable to the last survivor on the death of two of the bene- 
ficiaries and subtracting its present value from $25000, or we may 
treat the income as an annuity terminating on the second death and 
subtract its value from $25000. The remainder in the first instance 
would be the value of the annuity, and in the second instance, it would 
be the value of the insurance. Pursuing the latter course, we find the 
value of the annuity to be a Ky + a yz -\-a xz — 2a xyz . The values of which 
are 11.973 + 11.189 + 10.982-2x10.093 = 13.958, which multiplied 
by 1500 gives 820937 as the value of the annuity. The value of the 
remainder would then be 25000-20937 =4063.00. 

oo. Let it now be required to find the present value of the share 
or interest of A in the estate mentioned in the preceding problem. 

Solution: The problem may be divided so as to present two 
branches. First we may find the value of the interests of each in the 
annuity (income) in accordance with the principles employed in Ar- 
ticles 51 and 54. First to take A's interest. This will be one-half 
of an annuity on Ins life in connection with both B and C for the 
whole period of the annuity less the shares of B and C in the joint 
annuity on the lives of all three, that is, less two-thirds of the latter. 
That is, A receives \ l o(a xy +a xz ) — 2 /?.a xyz . The values are respectively 
11 . 973, 11 . 189 and 10 . 093. Solving, we get for his share in the annuity 
$7278.00 on an annuity of S1500.00. If we add to the value of the 
annuity thus found the value of his interest in the insurance as found 
at Art, 118 in the Chapter on Insurance, £2965.00, we will have 
$10243.00 as the value of A's interest, The other interests may be 
valued in the same way. 



CHAPTER X 
Of Life Insurance 

The following sections will be devoted to the discussion of methods 
and rules for computing the values of premiums, benefits, and so forth, 
of various forms of life insurance contracts. 

1. The Premium as applied to life insurance is the consideration 
paid or promised for the insurance. It may be paid in a gross sum 
called a single premium, at the beginning of the transaction, or, as 
is more common, in annual installments, continued through life, or 
through a specified term of years, called the Annual Premium. 

2. There is a distinction to be noted in premiums at the beginning 
of this discussion. The premiums written in the insurance contract 
are called the Gross or Office Premiums and may be greater or less 
than the mathematical value of the benefit promised in the policy 
though in practice they should be more. 

3. The premium which at the outset is of the same mathematical 
value as the insurance promised in the policy, the value of the premium 
and benefit being both computed on the same mortality table and at the 
same rate of interest, is called the Net Premium. Examples of net 
single premiums on an insurance of 1 at various rates of interest are 
given in Table No. XLIII. 

4. The premiums discussed here, unless otherwise stated, are net 
premiums and are based upon the value of the insurance contracted 
for. In an introduction to this part (Chapter 8), it was shown how the 
present value of an insurance is found. This value is the smallest 
sum for which the insurance can be sold without loss and from it the 
net annual premiums are derived, indeed it is called the net single 
premium. 

5. The net value of an insurance of $1000 on the life of a person 
aged 35, computed on Actuaries table and 4%, is $340.60. That is 
assuming the mortality rate shown by that table, $340.60, improved 
at 4% compound interest, it will amount to exactly $1000 on the 
average, at the end of the year in which the death of a person aged 
35 will occur. An insurance company may therefore sell an insurance 
of $1000 at that price and meet its obligation (leaving out of considera- 
tion the question of expenses). But it is seldom desirable to pay all 
this premium at once, and the companies are willing to accept a whole 
life or term annuity of equal value in lieu of the single premium. If 
we divide $340 . 60 by the value of an annuity due of $1 . 00 at the same 
age, and rate, and on the same table we will have a life annuity of value 
equal to the single premium. The value of an annuity due at age 35 
is 17, 144 and the annual premium, computed as last suggested, is 



COMPUTING ANNUAL PREMIUMS. 179 

$19.87. This latter annuity improved at 4% and accumulated 
during the life of a person aged 35 will also, on the average, provide 
the $1000 insurance at death. From the foregoing, is derived the 
formula for net annual whole life premiums : A x 

Px= (1) 

l+fl x 

P being the symbol of an annual premium. 

6. As an example, let it be required to find the net annual premium 
of an Ordinary Life policy of $1000 issued to A, aged 40 years, com- 
puted on the Actuaries table, and 4%. 

First Method: If we have tables of single premiums and life 
annuities such as tables 43 and 42, we simply apply the formula of 
Article 5, A 4 o 

P 40 = = 381 . 03^ 16 . 093 = $23 . 68, which is the required 

1 + a 40 
premium. 

Second Method: If we have a table of whole life annuities only, 
we may proceed on the following principles: A life annuity of 1 pays 1 
at the end of each whole year the life survives, but pays nothing the 
year of the death. Its value, as we have seen is o x . An insurance of 
1 pays 1 at the end of the year of the death. If we had an annuity 
which paid 1 each year the life begins it would include 1 for the year 
of the death, that is, the insurance. The value of such an annuity is 
obviously (1 + a x ), discounted one year or v(l + a x ). Now if we deduct 
o x from the latter value, all will be cancelled except the year of the 
death, which is the insurance. As a formula, this is, 

A x =v(l+ 0x )-o x . (2) 

A 40 = (16 . 093 X . 961538) - 15 . 093 = . 38103, which, multiplied by 1000 
gives $381.03. Again, this, if divided by (l+a x ) gives $23.68, as 
before. 

Third Method: If the insurance were 1 and payable at the begin- 
ning of the transaction its value of course, would be 1 but it is not 
payable until the end of the year of A's death. Hence, if A should 
pay his 1 down at the beginning, it would pay his insurance at the end, 
but he would lose the annual interest upon it during his whole life 
The value of this interest making it payable in advance is d(l-f a x ). 
Now if this value be deducted from the 1, the remainder will represent 
the value of the 1 payable at death, that is, of the insurance. Hence 
we have the formula A x = 1 -d(l + o x ) (3) 

A 40 = 1 - ( . 038462 X 16 . 093) = . 381032. Again this multiplied by $1000 
gives $381.03, which, being divided by 16.093, gives P =$23.68, as 
before. In this formula the character d is used to represent the rate 
of discount the interest rate being given. 

Fourth Method: Suppose we have only a table of single premiums 
from which to find the annual premium. In this case, we may suppose 



180 FINANCE AND LIFE INSURANCE. 

that A contracts his insurance of 1 but defers paying the premium until 
his death, when it is to be deducted from his insurance. He, of course, 
must pay interest on the single premium each year in advance, which 
is dA x . For this annual payment his beneficiary receives 1 — A x . 
Hence, dA x is the annual premium for an insurance of 1 — A x . And by 
proportion, P x :dA x : :1:1— A x , hence dA x 

Px= (4) 

1-A X 
P x = . 038462 X . 381032-f- (1 - . 381032) = . 02368. This multiplied by 
$1000 gives the annual premium, $23 . 68, as before. 

7. The four formulas just illustrated provide methods for finding 
the annual premium in any case where an annuity table or single 
premium table or both are available. They may be transformed 
algebraically so as to develop others almost as useful but the ones 
given are sufficient. 

8. Commutation columns afford the most convenient instruments 
for computing premiums, annuities and etc., and those required in 
this country are published in this book at those rates which are most 
usual. See tables 33 to 39, inclusive. 

9. The formula for single premiums has already been given, thus: 

M x (5) 

A x = 

D x 

A x A x 1 M x D x M x 

10. By Article 5, P x = = X = X-- = 



l+a x 1 l+a x D x N x -1 N x -1 
M x 

That is, P x = (6) 

N x -1 

11. Solving the problem of Article 6 by this formula we have, 
6242.4194-263643.62 = .02368 from which P 40 for $1000 can be found 
as before. 

12. If we have only the D and N columns, we may still find the 
annual premium by the formula D x 

d. (7) 

N x -1 

13. Solving the same problem by this formula, we have (16382 .558 
+ 263643 . 62) - . 038462 = . 062139 - . 038462 = . 023677, which, multi- 
plied by 1000, gives $23 . 68, as before. 

14. If it should happen that the computer has no data at all 
except a mortality table, or if the premium is desired for an insurance 
at a rate for which no auxiliary tables are available, resort may be 
had to Lubbock's formula, (Articles 16-18, Chapter III, title "Of 
Series") for finding the single premium. From this, the annual 
premium may be found by formula 4, Article 6, of this chapter. 



ANNUAL PREMIUMS, LIMITED PAYMENTS. 181 

15. The annual premiums just considered all relate to Ordinary 
Life Policies, that is, policies payable at death whenever it may- 
occur with equal annual premiums payable during the whole period 
of the life. It often happens, however, that all of the premiums are 
contracted to be paid during a limited term of years. Since the benefit 
is the same, its present value or net single premium is the same as in 
the ordinary life policy and the annual premiums for the limited 
premium-paying term must therefore be larger in order to be of equal 
value. It can not be impressed too strongly that whatever the form 
of the policy the present value of the benefit and the present value of 
the premiums contracted must balance at the inception of the contract. 

16. The net annual premium of a limited payment policy 

is found by dividing the net single premium by the value of a term 
annuity due on the same life, mortality table and rate of interest for 
the premium-paying period. As a formula this may be expressed thus : 
A x 

nP*.= — — (8) 

l+«xn-r-l| 

17. As an example, let us suppose that our policy already dis- 
cussed provides for annual premiums limited to ten years. Here, the 
single premium 381 . 03 must be converted into a ten-year term annuity 
with payments in advance. The 10-year annuity due at age 40 is 1 
plus a nine-year term annuity on the life of A. 381.03-i-8.0499 = 
$47.33 which is the annual premium required. 

18. By the commutation columns, the method is equally simple 
We have for the benefit side M x 

A x = 

D x 
and for the payment side, 
N x -i — N x+n -i 

. Dividing the former by the latter, we have, 

D x 

M x 
nP x = . (9) 

Nx-i — rsx + n-i 

19. Solving our problem by this formula, nP x = 6242.419-^- 
(263643 . 62 - 131765 . 62) = . 04733. Which, multiplied by 1000 gives 
$47 . 33, as before. 

The methods of computing single premiums and term annuities 
already presented in this and the preceding chapter are sufficient to 
enable the reader to compute the annual premium for any level premium 
limited payment life policy. Such policies are often called "Ten pay" 
or "Twenty pay", etc. "Life Policies." 

20. A policy in which the benefit promised is payable only in case 
of death within a limited period of time is called a Term Insurance 
Policy. The insurance guaranteed is called Term or Temporary 
Insurance. 



182 FINANCE AND LIFE INSURANCE. 

21. In the case of term insurance also, the benefit side and pay- 
ment side must, at the inception of the contract, exactly balance each 
other. This gives us the clue for finding the annual premiums. We 
first find the present value of the benefit and equate it against an 
annuity of equal value. 

22. First, to find the value of the benefit, the net single premium 
for term insurance, |nA x : If the term was for only one year, the value 

d x v 

of an insurance of 1 would be . If the insurance covered the second 

l x 

d x+1 v 2 
year only, its value would be and so on for the term and the value 

lx 

of all the n years, that is, |nA x would be: 

vd r +v 2 d x+ i + . . .v n d x +n 

(10) 

lx 

23. If both numerator and denominator of this series be multiplied 
by v x , the value will not be changed and the numerators will become 
d x v x+1 =C x ; d x+ iv x+2 =C x+ i;d x+2 v x+3 =C x+2 , etc., and the denominator 
D x , as we have already learned, and the series may be written: 

C x -f-Cjc+i -f-C x+ 2 "f C x + a 

. The numerator is that portion of the summa- 

D x 

tion of C which produces M, which extends to M x+n . Hence we have 
the formula, M x — M x+n 

|nA x = (11) 

Dx 

24. Formula 10 of Article 22 will enable the computer to get 
|nA x by means of the mortality table alone with little labor if the 
term is short, but if he has a long term an approximate value may be 
found by the formula for finding the nth term, which has been suffi- 
ciently explained. Or, he may employ Lubbock's Formula computing 
a sufficient number of equidistant values of one year insurances to 
give at least second differences of both u o and U m. The formula for 
this purpose is as follows: 

n-1 

|nA x =n(u +Ux + . . +u m -iH (u m -u )- 

2 
n 2 -l n 2 -l(A 2 um- A 2 u )^- etc. 

(Aum-Au )+- (12) 

12n 24n 

25. Having found the net single premium for a temporary insur- 
ance, we have but to find as the annual premium, a term annuity of 
equal value, which is done by dividing the single premium by the 



ANNUAL PREMIUMS, DEFERRED INSURANCES. 183 

value of a term annuity due of 1 for the term during which the annual 
premiums are payable. The formula is, 

|nA x 
|nP x = — - (13) 

l+«x n-l| 

26. By substituting for the numerator and denominator of the 
last number of equation 13, their equivalents from the commutation 
columns, and simplifying the result, we have : 

M x -M x+n 

|nP x = (14) 

N x -i-Nx +n -l 

27. A Deferred Insurance is one which does not begin until 
after the lapse of some period of time. Its value at the time the risk 
begins would be computed as an ordinary life policy at the deferred 
age, but this value would not represent the present value or net 
single premium for the insurance, because it is not now due and must 
be discounted to the present date and because further, the insured may 
die during the intervening term, and liability never attach. The value 
must be reduced also by this probability. From this we see that the 
present value or single premium may be found by first finding the value 
of the insurance at the time it begins, A x+n , and multiplying this by 

V n l x+n 

or v n np x . That is, n|A x =A x + n v n np x . (15) 

1* 

28. By the commutation columns, the equivalent formula is 

Mx + B 

n|A x = (16) 

D x 

29. If the annual premiums are payable during the whole life, on 
principles already explained, the annual premium would be, 

n|A x 

n|P x = - (17) 

l+a x 

30. The equivalent of the last member of equation 17 may be 
stated in commutation symbols by substituting the formulas for the 
numerator and denominator and simplifying the result whence we have 

M x+n 

n|P x = (18) 

N x -i 

31. If the premiums are limited to any number of, say, n years, 
we have: 

n|A x 

nPn|A x = — (19) 

l+axn-l| 



184 FINANCE AND LIFE INSURANCE, 

32. The equivalent of the last formula stated in commutation 
symbols is M x+n 

nPn | A x = _ (20) 

N x -i-Nx + n-l| 

33. It may be remarked also that the value of a deferred insurance 
may be found be deducting from the value of a whole life insurance 
the value of a temporary insurance for the term intervening between 
the date of the insurance and the beginning of the risk. The following 
equations hold: 



A x =n 
A x — n 



A x+ |nA x ; 

A x = |nA x and A x — |nA x =n|A x . 



Also that the approximate formulas applicable to whole life insurances 
may be used in computing the values of deferred insurances, making 
the proper discounts for present value and probability of living as 
explained in Article 27. 

34. It may be helpful to give in this section, numerical examples 
illustrating the use of each of the last eleven formulas. Using our 
same $1000 policy on A. 

vd x +v 2 d x+1 + . . v n d x+n 
(10). nA x = Let n =5 years. 

lx 

v. dx. lx Ax 

J . .961538x815=783.647-^78653 = .00996. 

2 . . 924556 X826 = 763 . 683^- 78653 = . 00971. 

3 . . 888996 X 839 = 746 . 8684- 78653 = . 00949 

4 . . 854804 X857 = 732 . 5674- 78653 = . 00932. 

5. .821927X881 =724. 118^78653 = .00921. 



3750.8834-78653 .04769 

The last sum multiplied by 1000 gives $47.69 as the net single 
premium for an insurance on A, aged 40, for a term of five years, by 
Actuaries table and 4 per cent. The first fine by the figures at the 
right shows the present value of an insurance for one year. The second 
line the present value of an insurance for one year deferred one year. 
The third line the present value of an insurance for one year deferred 
two years, and so on. The method is of not much practical value 
but shows lucidly the principle underlying life insurance, for by it the 
value of an insurance for a whole life, or any part of a fife, may be 
computed. If a term insurance is desired, the sum of the insurances 
for the years of the desired term will be the value; if a deferred insurance 
is desired, the sum of the years from the time the risk begins is the 
value ; if a deferred term insurance is desired, it is only necessary to 
follow along the years until those comprising the term are reached and 
their sum will be the value. 



INSURANCES, PROBLEMS AND SOLUTIONS. 185 

M x -M I+n 

(11) nA x = Let n=20 years. 

D x 
Then, (6242.42 -3189. 47)-M6382. 56 =$186.32. 
The last value is the net single premium for a 20-year term insurance 
on the same life, table and rate. 

(12) Equation 12 will be illustrated in connection with joint life 
insurances, which see. 

|nA x 

(13) |nP x = Let n=20 years. 

l+Ox: n -l 

Then |nA x =$186.32. c xl 7| =11.710 and we, by substitution, have 

186.32 
|20P 40 = =$14.66. 

12.710 

That is, $14.66 is the net annual premium for a 20-year term policy 
on the life of our subject A, by Actuaries table and 4%, the premiums 
to be paid annually during the term. 

M x -M x +n 

(14) |nP x = . Let n again be 20 years and from the 

X x -i-N x+n -i 

Commutation table, we have as the second member of the equation 
(6242.42-3189.47)-K16382.56-5320.82) =$14. 66. 

(15) n|A x =A x+n v n np x . Let n again be 20 years. A x+n =A 60 = 
599.43; v 20 = .456387 and 20p 40 =l 6 o = 55973-M 40 = .78653 and we have 
599. 43 X. 456387 X. 7 1160 = $194. 68. The latter sum then is the net 
single premium for $1000 insurance on the life of A deferred 20 years. 

Note: By article 33 supra the sum of this and the 20-year term 
found above, $186.32, should equal the whole life premium on A which 
we have found is $381.03, and so it does. 
M x + n 

(16) n[A x = . Let n again be 20 years. M 60 =3189.47, 

D x 

D 40 = 16382.56, then 3189.47 

= .194687, which multiplied by 1000 

16382.56 
gives $194.69. 

n|A x 194.69 

(17) n|P x = . Let n again be 20 years. n|P x = =$12.10. 

l + o x 16.093 

The latter sum is the annual premium for an insurance on the life of A, 
deferred twenty years, the premiums being payable during life. 

M* + „ 

(18) n|P x = . Let n be 20 years, as before, then 

N x -, 



186 FINANCE AND LIFE INSURANCE. 

3189.47 

20P 40 = =.0120976, which multiplied by 1000 gives S12.10 

263643.62 
as before. 

n|A x 

(19) nPn|A x = . Let n be 20 years as before, then 

1 + Ox n — l 

194.69 

20P 2 o|A4o= =$15.32, which is the annual premium for the 

12 71 

deferred insurance of the last problem, the premium-paying period 
being limited to twenty years. 

M xfn 

(20) nPn|A x = . Let n again be 20 years, then from 

the commutation table, we have, 

3189.47 

= .01532, from which the premium for 1000 may 

263643.62-55414.91 

be found by simple multiplication as before. 

(35.) We may again assume that cases may arise when auxiliary 
tables or some of them are not at hand, and show how the premiums 
may be found. Thus, suppose we have only the N and D columns. 
For a Term Insurance the following formula has been developed for 
the annual premium: 

D x -D x+n 
|nP x = d. (21) 

Nx-l— Nr + n-l 

Taking the example already used, we have 

16382.56-5320.82 

20P 40 =— .038462 = .014661, 

263643.62-55414.91 

which multiplied by 1000, gives the annual premium, $14 . 66. 
(36) Employing the N column alone, this formula may be used: 

N x -N x+n 
|nP x =v (22) 

N x -l-N x + n -l 

These values taken from the tables are 



/ 247261 .06 -50094. 09 \ 
\263643.62-55414.91/ 



.961538-1 I = .014661. 

\263643.62-55414.91/ 

From which the annual premium for 1000 is derived as before. 



NATURAL PREMIUMS, FRACTIONAL PREMIUMS. 187 

37. In some forms of policy, the consideration is made what is 
called a Natural Premium. That is, the annual cost of each year's 
insurance is charged. This cost increases with the age of the policy 
holder. In the example supposed, the simplest way of finding these 
premiums would be to look into that column of the mortality table 
showing the probability of dying each year and follow down to age 40. 
It will then appear that the probability of a person aged 40 dying 
during one year is .010362. From which we learn that 10.36 persons 
out of 1000 will die during the year of that age, requiring $10362.00 
to pay each of the beneficiaries of the deceased persons $1000, which 
will be $10.36 for each of the 1000 insured persons. The latter sum 
is based on the further supposition that it is either paid at the end of 
the year or that money bears no interest. If paid in advance as is 
usual, and it bears interest at 4%, the premium would be 10.36-v- 1 .04, 
or 10. 36 X. 961538 (v) =$9.96. That is, it would only be necessary 
to pay the present value of the annual cost. By the same table, we 
find the next year's cost to be $10.61, the 10th year, $15.94, and so on. 

38. Mr. Abb Landis, in his recent book, Life Insurance, has taken 
the pains to demonstrate that the natural premium plan of insurance 
has many advantages for those who desire insurance, not as an in- 
vestment, but for the protection of dependents. 

39. It is sometimes desirable to make premiums payable in semi- 
annual, quarterly, monthly or other fractional parts of a year. Indeed, 
in the case of what is called Industrial Insurance, the payments are 
often due weekly or bi-weekly. And in the case of Fraternal Insur- 
ance, the monthly payment is almost universal. The value of these 
payments are of course susceptible of accurate computation, but in 
practice, less intricate methods, producing approximately correct 
results are employed. Some of these will now be discussed and il- 
lustrated. 

40. We have seen that the value of the premiums, however paid, 
must equal the value of the insurance. In other words, the payment 
side of the contract must equal the benefit side. Ordinarily, the whole 
annual premium is payable in advance, and consequently may be at 
once placed at interest by the company. If only an installment 
(fraction) of the annual premium is paid in advance, the interest on 
the remainder is lost until it falls due, the amount depending on the 
number of installments. The formula for the annual premium has 
already been given and is A x 

P x = . Also P x (l + o x ) =A X . But in case 

l + a x 

of fractional premiums instead of multiplying P x by an annuity due of 
1, we have for the factor a fraction of 1 due. Thus P^™ ) — 






and for the value of the premiums for the year, we have 



188 FINANCE AND LIFE INSURANCE. 

A x 

P x = » in which m represents the number of installments 

1/m + a™ 
into which the annual premium is divided. 

We have seen (Annuities, Art. 50) that an approximate formula 

m-1 

f or a x is a x — ttx _| } an( j the above formula may be written : 

2m 
A x 
* x = • Simplifying the right member of this equation, 

it becomes, 

A x 

P ( ?= (23) 

m + l . 

41. Taking our familiar example, but assuming the premiums 
for the year to be made payable in four installments, we have the 
following result: 381.03-^[ ( 5 /s + 15. 093) =(.625+15.093) ] =24.40, 
the quarterly payment being $6.10. 

42. Suppose that the installments are payable monthly. The 
factors then become 381.03-^ ( 13 /-24 + 15.903) and the value of the 
annual premium is 24 . 36. Hence, the monthly payment is 2 . 03. 

43. It is said to be customary to add 4% to the annual rate and 
divide the result by 2 to obtain the semi-annual premium, and to add 
6% and divide by 4 for the quarterly premium. By this method, the 
quarterly premium would be $6 . 35 instead of $6 . 10. The excess 
charge is probably warranted by the fact that in practice, there is 
considerable expense incident to collecting, two, four or twelve payments 
per year instead of one, besides the loss of interest on the deferred 
installments. 

44. When commutation columns computed on the monthly basis 
are available, the monthly payments for whole life, term and deferred 
insurances may be found by the formulas already given, dividing the 
result by 12. The formulas assuming the following form: 

M (1 X 2) 

P ( x 2) = . (24) 

12N (1 X 2 >-, 

M (1 |>-M (1 X 2 \ D 
1^)= _ (25) 

12(N (1 x 2) - 1 -N( <1 x 2) + D - l) 
M (1 x 2) + » 

n|P (, x 2, =- (26) 

12N <12) -. 



ENDOWMENTS AND ENDOWMENT INSURANCES. 189 

45. Ordinary commutation columns may be converted into col- 
umns on the monthly basis, to which the foregoing monthly premium 
formulas and others are applicable. Mr Landis suggests the following 
approximate formulas for making the conversion: 

D (1 X 2) = V 2 (D X +D X+1 ) (27) 

N (1 X 2) =2D (1 X 2) (28) 

C (1 x 2) = (l+V2i)C x (29) 

M (1 X 2) =-C (1 X 2) (30) 

46. The Greek letter 2 will be recognized as the symbol for 
summation, indicating that D x and C x are to be summed in the 
manner with which the reader is already familiar. 

47. In the foregoing sections, except 43, the word installment has 
not been quite accurately used to describe the premiums for fractional 
parts of a year as they are not installments of the regular annual 
premium, but are approximate net premiums for the shorter periods, 
derived from the value of the benefit. 

48. A form of insurance called Installment Insurance, Pension 
Policies, and other names, has come into use. Some of these are mere 
life or temporary life annuities, or annuities certain engrafted upon an 
endowment insurance or limited pay life insurance, by the terms of 
which the value of the policy at the settlement period is employed to 
purchase the annuity. That is, at settlement the value is paid in annual 
or other installments instead of at once. In other cases, the contract 
consists of a deferred annuity. In many cases, the benefit payable 
at the death of the insured is made payable in installments at the 
option of the insured. In such case, the face of the policy is usually 
converted into an annuity certain or a temporary life annuity on the 
life of the insured, of equal value. The sum which at the settlement 
period is employed to purchase the annuity with any intervening 
benefit may be treated as the benefit side of the policy, and the premium 
or payment side determined therefrom in the usual way. 

49. An Endowment Insurance is one which combines a pure 
endowment with an insurance covering the term intervening between 
the date of the contract and the maturity of the endowment. 

An Endowment or Pure Endowment within the meaning of 
this section is a sum to be paid to a certain person called the nominee, 
in case he shall be alive at a certain time. 

50. As an example of a pure endowment, let us assume that our 
familiar subject is to receive his $1000 if he be alive at the end of ten 
years. What would be the present value, or single premium, for 
this benefit? 

First, it is not due in any event for ten years and hence must be 
discounted at the rate of four per cent for that time. That is v 10 . 



190 FINANCE AND LIFE INSURANCE. 

Second, he may not survive the term of ten years and stands this 
chanee of losing the benefit. This chance we have learned is np x , in 
this case, 10p 4 o, or the probability of living ten years at age 40. This 
value is the same as the present value of the 10th payment of an annuity 

V 10 l 40+ 10 V n l x +n 

and is or in general terms, , Its symbol is E x and the 

I40 lx 

formula for a pure endowment is therefore, 

D x + n 

E X n| =v n np x or E xn | = (31) 

D x 

51. Numerically, the solution is as follows: E 4 o io| = (69804 X 
.675564-^78106) X 1000 =603.76. Or by formula 31, (9822. 30-f- 
16268.62) X 1000 =603. 76. 

52. The value, single premium, for an endowment insurance is 
|nAE x and must equal the present value of two benefits. First, of a 
pure endowment, payable at the end of the term if the insured survives, 

Dx + n M x -M x+n 
and second of a temporary insurance for n years, 



D x D x 

The whole present value may then be expressed in the formula 

M x -M x+n +Dx+n 

|nAE x = . (32) 

D x 

The symbol for this insurance is also written, A X n|. 

53. Solving our same problem by the last formula, the numerical 
work is as follows : (5979 . 94 - 4586 . 90 +9822 . 30) + 16268 . 62 X 1000 = 
689.39. 

As illustrating the explanation given in Art. 52, we may find the 
value of the insurance alone, thus, (5979. 94 -4586. 90 )-M 6268. 62 = 
85 . 63, the value of the insurance feature. This deducted from 689 . 39, 
gives 603.76, the endowment feature as found in Article 51. The 
principle of formula 2 also applies to endowment insurances and we 
have, A X n| =v(l -\-a x ~i\) — (l^i\. Also Formula (3) may be employed 
taking the form, A xn | = 1 — d(l + a xn -i|). 

54. The formula for the annual premium then on principles al- 
ready elucidated is, M x — M x+n +D x + n 

P«| = (33) 

Nx-l— Nx + n-l 

55. As an example of the last article, we may find the annual 
premium of an endowment insurance on the life of our familiar example, 
making the term ten years, when we have the figures: (5979.94 — 
45869 +9822 . 30) + (267505 , 97 - 136120 , 05) X 1000 - 85 . 36. 



SEMI ENDOWMENTS— FORMULAS AND SOLUTIONS. 191 

56. If we have the single premium and term annuity computed, 
the annual premium is readily found by dividing the single premium 
by the value of an annuity due of 1 for the payment term of the endow- 
ment, the formula being 

|nAE x 
Px" n | = — (34) 

1+Cxn-l 

57. To find the annual premium in the problem of Article 55 by 
the last formula, we have 689 . 39 

P 4 o i"o| = =85.36. 

1+7.077 

58. The endowment feature of an endowment insurance policy 
does not always coincide in amount with the insurance feature. Thus 
it is not unusual in this country, and is said to be of frequent occurrence 
in England, to write policies called semi-endowment policies. In 
such a policy, the temporary insurance is twice the amount of the 
endowment. In such cases, the two features of formulas 32 and 33 
would be treated separately and the results combined or the endowment 
branch could be divided by two before combining the parts. The 
formula will then assume the form, 

2(M x -M x +n)+D x+n 

(35) 

2D X 

If the problem of Article 53 be so changed that the endowment 
to be paid is only 500, the temporary insurance remaining 1000, the 
solution is as follows: [2 (5979. 94 -4586. 90) +9822. 30 ]-f- (2 X 

16268.62) X 1000 =387.51. If to one-half the value of the pure en- 
dowment found in Article 51, that is, of 603.76, we add the single 
premium of the temporary insurance found in Article 53, that is, 
85.63, we have 387.51 as above, proving the correctness of the com- 
putations and also illustrating the nature of such contracts. It may 
be remarked that the benefits of endowment and insurance contracts 
may be, and in practice often are, of various natures, with premiums 
payable in various ways. The problems thus presented need only to 
be analyzed and understood, and the solution, usually will be obvious 
enough. For instance, the policy may provide for a pure endowment 
with clauses insuring only the annual premiums, or the temporary 
insurance may be one-half or some other proportion of the endowment. 
Again, the premiums may vary in amount, temporarily or throughout 
the term. Some of these contracts will now be considered. 

60. Suppose the endowment is 1000 and the temporary insurance 

M x -M x + n 

500, that is, one-half the endowment. The insurance is and 

2D X 



192 FINANCE AND LIFE INSURANCE. 

D x + n 

the endowment . Reducing to a common denominator and 

D x 

M x -M x+n +2D x + n 

adding, we have the whole benefit, 

2D X 

61. If the annual premium for the last benefit be required, we 
may substitute N x -i— N x + n -i for D x and the form of the equation 

M x -M x+n +2D x + n 

becomes: nP x : n | = (35a) 

2(N x - 1 -N x+ n-i) 

62. Formula 35 may be converted into a formula for the annual 
premium in the same way. 

63. Suppose the annual premiums are a certain sum, say for five 
years and twice said amount for the remainder of life. What is the 
net annual premium? In this, as in all cases, the benefit side and 
payment side of the contract must be equal. The value of the insuranc 9 
is A x =P[(l + a x ) +4J a x ] or in commutation columns, we may write, 
the benefit side, M x and the payment side, P x (N x -i+N x+5 -i), from 
which we have the formula, (using x as the symbol for a special premium 

M x 
x.= (36) 

64. Supposing our usual subject A to contract such an insurance, 
we have the numerical result, 5979.94-^-267505.97 + 193619.95 = 
. 01297 X 1000 = 12 . 97 = x and 2x,25 . 94. The level net annual premium 
for age 40 by American Experience table and 4% is 22.35 per 1000 and 
at age 45, by the same standard the net annual premium is 27 . 12. 

65. As another example illustrating this class of insurance, let it 
be required to find the net annual premium of a policy of $1000 issued 
to A aged 43 years, the gross annual premium being $18.27 for five 
years and 35 . 40 for the rest of life, computed on Actuaries table and 
4%. The ratio between the initial and final premium instead of being 
1 to 2 is 1 to 1 . 938. Turning now to the commutation columns, we have 
M 43 =5764. 77, N 42 = 216841. 25, N 47 = 152991.67. The latter multiplied 
by . 938 gives 143506 . 19. Adding the last number to N 42 , we have for 
the divisor, 360347.44. Performing the division and multiplying by 
1000, we have x =16.00 and for the life premium 16.00X1.9376 = 
31 . 00. The annual net level premium at age 43, by this table, is 26 . 58 
and at age 48, 32.77. The single premium at 43 is 408.71 and the 
value of these premiums must equal the latter sum. A life annuity 
due (l + a x ) at age 43 is 15.374). This multiplied by 16.00 =245.984. 
A life annuity due, deferred five years at age 43, is 10 . 847 and this 
multiplied by 15.00 = 162.705. 

245.984 + 162.705 gives the single premium, 408.69. 



VARYING PREMIUMS, EXAMPLES. 193 

66. The principle illustrated in the last article will enable the 
computer to solve problems when several increases in the premium are 
to be made at stated intervals. The annual premiums begin with a 
sum which ma}' be regarded as unity and to this is added deferred 
annuities due entered upon at the periods of increase. The sum of 
these divided into the value of the insurance gives the initial premium 
from which the others may be derived. The same principle is employed 
in the solution by means of the commutation columns. 

A more general formula for the special net premium discussed in 
the last article is as follows: 

M* 

x- (37) 

N x -i^=k(N x+ t-i +N x+2 t-i&c =N x+ mt-l 

in which k represents the ratio of the increase (or decrease) in the 
premiums, t represents the periods between the changes in the prem- 
iums and m represents the whole number of variations which take 
place in the premiums. 

67. If the premiums are unknown, that is, it if is desired to find 
an appropriate step rate as compensation for a given insurance 
benefit, the following formula has been developed: 

k(N x +t-i +N x+ 2t-l +N x+ 3t-i &c +N x+ mt-l) 
t=M x =f (38) 

N x -! 

68. As a numerical example of the last formula, let us take one 
from Mr. Landis' book, Life Insurance: Assume that the insurance is 
1000 for whole life and that the benefit is uniform and taken at age 35, 
and that the contribution is P', and increased by 2 for three years and 
then continues uniform for the rest of life. 

The benefit side 1000M 35 , 

The payment side =P'N 35 +2N 3 6+2N 3 7+2N 38 , 

Transposing so as to let P'N 3 5 stand alone, and dividing both members 

1000M 35 - (2N 36 +2N 37 +2N 38 ) 
by N 35 we have P' 3 s = = 

N 3b 

7049683 . 60 -2(400925 . 92 +378594 . 17 +357256 . 75) 

424294.62 
7049683 .60-1 136776 . 84 



11.25 



424294 . 62 



The values used are taken from N. F. C. tables at 4% published in this 
book, where, according to American usage N x =N x -i of tables arranged 
in accordance with the recommendation of the International Congress 



194 FINANCE AND LIFE INSURANCE. 

of Actuaries. The N columns of that table have been rearranged in the 
present work to conform to the Congress rule followed in the other 
tables. The initial rate above found pays the first premium. The 
second, by the problem is 13.25, the third 15.25 and the last 17.25, 
which remains level throughout the life. 

69. It occasionally happens that an insurance contract provides 
for a Varying Benefit in one respect or another. That is, the amount 
paid at death differs at different periods in the life of the policy. More 
frequently this occurs where the annual premiums as well as the face 
of the policy are paid in case of death during the premium-paying 
period. In some instances, annual additions, or additions to be made 
at a settlement period or at death are provided for. Some of these 
features may be treated by means of the R. commutation columns. 
Others have to be treated by special methods which are usually indi- 
cated by the nature of the problem. Some examples will best develop 
the methods of solving these problems. 

70. An Increasing Insurance, the symbol of which is IA X is 
one which increases by some law usually by its own amount. Thus, 
by commencing at 1 and increasing by 1 each year. In the case of 
insured annual premiums, the initial insurance would be P increased 
by P each year. In the first case, the insurance provided expressed 
in commutation columns would be 

M x M x+ i M x+2 

1 1 etc., to end of table. 

D x D x D x 

That is, we have an insurance of 1 for life, to which is added the next 
year a deferred insurance of 1 for life and so on. If we sum the M 
values beginning with the oldest age and continuing to the youngest, 
recording the sums, we will have a column which is called R, which 
would have the quality of giving the value, single premium, for an 
increasing insurance beginning at the age taken and extending through 
life, when divided by the D for the corresponding initial age. Thus 

we have the formula R x 

IA X = (40) 

D x 

71. Suppose a policy has for its consideration 20 equal annual 
premiums of 37 . 70 which are to be returned in case of death during the 
premium-paying period. What is their present value or net single 
premium, the insured being 40 years of age, computing by Actuaries 
table and 4%? If we write the number R x — R x+n as the numerator 
just as in case of an ordinary temporary insurance, it is clear that the 
result will not be correct because 20 insurances of 1 will still remain 
uncancelled and it will therefore be necessary to deduct 20M 6 o to cancel 
these. We therefore have the formula: 



INCREASING INSURANCES, FORMULAS AND SOLUTIONS. 195 

R40 — Reo — 20Mdo 



IA40 20 1 



which is numerically (133639.96-37833.97-20 X3189.47)-i-16382.56 
= 1 . 9543. The latter sum must be multiplied by 37 . 70 which gives 
the single premium sought, 73.68. 

72. Let us now find the net single premium for the whole benefit 
of the insurance of Article 71, including the return of the premiums 
in case of death during the 20 years premium-paying period. We must 
have 1000 (M 40 -f-D 40 ) + the value of the return premises as already 

found. This is 10C0 M 40 37 . 70 (R 40 - Reo - 20M 60 ) 

+ (42) 

D 4 o D 4 o 

or [1000 M 4 o+37.70(R 40 -R6o-20M C o)]-^D 40 . The values of M and 
R may be substituted from the table and the result obtained by the 
arithmetical processes indicated by the signs, will be $455 . 32. 

73. The net annual premium for the policy will be found by 
dividing the last sum by a 20-year annuity due on the same life, 12. 71 
giving as the net annual premium, $35.82. 

74. The symbol and formula for an insurance of a certain sum 
increasing (or decreasing) each year by another sum and thence remain- 
ing constant during the remainder of life are sometimes written as 
follows : 

kM x ±h(R x+ i-R I + n ) 

d'nlAO = (43) 

D x 

In this formula, k represents the initial sum insured and h the annual 
increase. It differs from the preceding formula in that instead of 
cutting the insurance down to the initial sum after n years, only the 
future increase is cut off the amount of insurance remaining at the 
point reached at age x+n. 

75. For an ordinary life policy with return of the net annual 
premium, along with the capital benefit, the formula may be written 
as follows: 

M x 

* = (44) 



Xx-X-Rx 

76. The benefit secured by the policy of the last article is A*, being 
the value of the capital insurance and x(IA x ), that is an increasing 
insurance of 1 multiplied by the net premium, %. The net annual 
premium would be found by dividing the sum of these values by 1 + a x . 
The equivalent of these is, in terms of the commutation columns, 



196 FINANCE AND LIFE INSURANCE. 

M x + xR x Nx-i M x + xR x 



X 



D x D x Nx-i 



Multiplying by N x -i, we have 

xNx-i =M x+ xR x . Transposing, we have 

x(N x -i — R x ) =M X and dividing by the co-efficient of x we have finally, 

M x 

x = , which is the formula assumed in Article 75. 

Nx-i-Rx 

77. The more usual form is to return the gross annual premiums, 
but it is sometimes important to find the net annual premium of such a 
policy for the purpose of valuation. The formula for an ordinary life in- 
surance with return of gross premiums may be written as follows : 

M X +P'R X 

x= (45) 

N x -i 

In this formula, P' is the symbol used for a gross or contract premium. 

78. The reasoning on the last form of policy is similar to all in 
that the benefit side and premium side of the contract must balance 
at the inception of the contract. That is 

x(l + a x ) = A x +P / (IA x ), which, in commutation column terms, is 

Nx-i M x P ; R X 

x = H from which we have 

Dx D x D x 

Mx+P'Rx Nx-i Mx+P'Rx 



D x D x Nxi 

In applying formula 45, M x must be given as a eo-efficient, the sum on 
which the gross annual premium is based and the co-efficient of R x is 
the gross premium. 

79. As an example of the last formula, let it be required to find the 
net annual premium of an ordinary life policy covering both policy and 
premiums, the policy being for 1000, on the life of A, aged 40 years, 
the gross annual premium being 53 . 24 employing Actuaries table and 
4% in the calculation. From the table and formula we get (6242 . 42 X 
1000) +133639 ..96X53. 24-^263643. 62=50. 67. The latter is not a 
well balanced premium, the annual gross premium having been fixed 
arbitrarily. 

90. Let us suppose the loading of the premium of the last policy 
to be 25% of the net premium plus a constant addition of 5 per 1000 
or, .005 on 1. Representing the percentage by k, the constant by c, 
and the gross premium by P' =P(l+k)+c and for the special annual 
premium, the formula: 



SPECIAL PREMIUMS, LOADED PREMIUMS. 197 

M x +cR x 



Nx-i-(l+k)R, 



(46) 



The solution is as follows: (M 40 = 6242. 42 +R 40 = 133639.96 X .005)^ 
N x -i=263643. 62 -133639. 96 XI. 25 =6910. 61^96589. 10 = .07155 
which, multiplied by 1000 gives 71 .55, the net annual premium sought. 
The gross premium would be found by multiplying 71 .55 by 1 .25 and 
adding 5. as indicated by the formula P(l +k) -f-C. 

91. A more usual form of policy is one having limited premiums 
all or part of which are returned in case of death during the premium- 
paying period. An example and some explanation of one form of the 
problem presented by such insurances is given in Articles 71 to 74. In 
that case, an arbitrary gross premium was assumed. Suppose that in 
the last problem considered, we substitute n annual premiums, the 
same to be returned in case of death during the premium-paying 
period, the loadings to be as before. The value of the benefit is first 
A x , the single premium for whole insurance, second x(IA)x' n| an in- 
creasing term insurance covering the gross annual premiums, which 
may be written (ic(l +k) +c) (IA)x / n|. To obtain the net annual 
premium, the value of the sum of these benefits will be divided by the 
value of an n year annuity due. l + Oxn-i|. Writing these values in- 
terms of the commutation columns and simplifying algebraically, the 
following formula is developed: 

M x +c(R x -R x+n -nM x+n ) 
% = (47) 

N x -1 -N x+n -l - (1 +k) (R x -R x + n-nM s + n ) 

92. In case the conditions of the policy were the same as in Article 
91, except that the premiums are to be repaid along with the capital 
benefit at death whenever it may occur, the formula assumes the 
following form: 

M x +c(R x -R x + n ) 

tc = (48) 

Nx-i-Nx+n-i--(l+k) (R x -R x+n ) 

Note: In the last two articles the gross or loaded premium 
would be P' = x(l +k) +c. It should be further noted that the difference 
in the two last equations is that only R x+n is cut off of the increasing 
insurance in the latter, leaving the n premiums paid uncancelled. In 
the former equation, the whole premium benefit is cancelled at the end 
of n years by deducting n(M x + n ). 

93. The principles illustrated in the preceding Articles will enable 
the reader to deal intelligently with the more usual forms of varying 
benefits. It may be remarked that the commutation columns are 
adapted as well to the decreasing insurances and annuities as to in- 
creasing ones, but there is in practice little occasion for such application. 



191 FINANCE AND LIFE INSURANCE. 

94. A Tontine Fund represents the accumulations of annuities 
granted at some previous time, but not drawn, and is to be distributed 
among the survivors of the group. The annuities are in other words, 
forborne during the tontine period. The formula given for the share 
of a given annuitant, is: 

N x - a -N x 

-u\a x (49) 

D x 

95. A similar principle is involved in the formula: 

Mx-n-Mx 

|-nAx= (50) 

Dx 
which represents the value of the accumulated annual costs of insurance 
including interest during the term of n years, on the supposition that 
said annual costs are forborne or not paid. 

96. All of the insurances discussed in the foregoing articles are 
payable at the end of the year in which the death of the insured shall 
occur. On the assumption that deaths at a given age are uniformly dis- 
tributed throughout the year, the payments are due six months after 
death. As a matter of fact, deaths do not occur thus uniformly, but 
the increase in the force of mortality for so short a period is negligible 
and is generally disregarded in practice. 

97. If it is desired to make the insurance payable immediately 
after death, the company should be compensated for the loss of six 
months interest which it would sustain by the earlier payment. That 
is, it should receive roughly, AxG+J). or, more accurately, A x (i+0* 
and for such an insurance, also called a continuous insurance, we have 
the formula: Ax = A x (i+i)*. For an insurance payable three months 
after death, the approximate value would be A x (1+4) and for one month, 
Ax(i+i2) The value of A x may also be found by the formula, A x = 
1— d'ax- Also, conversely, A x = Ax(i + i) — *. Here d' represents the 
force of discount. 

98. Joint Life Insurances are governed by principles similar in 
most respects to those controlling insurances on single lives, and most 
of the formulas considered in this chapter may be employed to com- 
pute premiums for such insurances by merely substituting the symbols 
for joint lives in the formula and their values in the equations, thus: 

(51) 
(52) 

(53) 

(54) 
1-A xy 



09. 


Axy=V(l+Oxy)-«xy 


100. 


Axy = l-d(l+a xy ). 




Axy 


101. 


P = 


•fxy = 




1-fflxy 




dAxy 


102. 


Pxy 



JOINT LIFE FORMULAS. 199 

M x , 

103. A iy = (55) 

D xy 

M ry 

104. P iy = (56) 

Nx-T.y-1 

105. The computation of joint life commutation tables for all 
combinations of three or even two lives is an immense undertaking and 
recourse must be had to methods of approximation. For all kinds of 
life tables, Lubbock's formula is probably the best since it does not in- 
volve as one of its factors the force of mortality. Equidistant values 
of single year insurances for use in the formula may be computed by 
the formula: 

V n (lx + n: y + n ~~ lx + n+l^y + n+l) 



in which n is general representing any number of years. For insurances, 
the form of the formula is : 

n-1 n-1 

A xy =n(uo+u a +u,a+U3n + ) -JoH -Auo-etc. 

2 12n 

See Articles 54 and 55, Chapter 4. 

But it is obviously less laborious to use either formula 51 or 52 in 
the calculation of the single premium which requires the approximate 
value of the annuity only and the same annuity may be employed to 
find the annual premium, thus, suppose the value of a joint life annuity 
on three lives aged 30, 36 and 41 is 11.375. Now, by formula 51 
A xy , = v(l+a xyz )-o xy2 = (.966184X12.375) - 11.375 = 11.957- 
11.375 = .572, which is the single premium. Also, 

A ry8 
p xyi = and 

l+Oiy« 

therefore, P xyz = .572-r- 12.375 = .04703, which is the annual prem- 
ium. For 1000, it, of course, would be 47.03. 

106. Illustrating the computation of single and annual premiums 
by means of Lubbock's Formula, let it be required to find the single 
and annual net premiums for an insurance of $1000 on the life of A, 
aged 35, and B aged 45, using American Experience and 4%. We may 
use formula 51 and here the value of a joint life annuity is the first 
requisite. First compute the present value of six payments in 10, 20, 
30, 40, 50 and 60 years which will carry us to the end of the table. To 
do this, four columns may be prepared. In the first, write the logar- 
ithms of the present value of 1 at four per cent in 0, 10, 20, 30, 40, 50 
and 60 years which will carry the younger of the two lives to the end 
of the table. In the second column, write the logs of I35, I45, I55, Us, 
1 7 6, las and I95 which carries that life to the end of the table. In the 
third column, write logs of I45, U& and etc., to 1 96 . Add the logs of 1 3S 



200 



FINANCE AND LIFE INSURANCE, 



and I45 and deduct the sum from 10 — 10, that is, take the co-log or 
arithmetical complement of the sum and place it in the fourth column. 
Add the four columns across the page forming the 5th column which 
will be the logs of the payments. Take out the natural numbers, and 
these will be the payments for the 1st. 10th, 20th, &c. years. Arrange 
these in a column for differencing. Take out to third differences and 
sum by the formula. The work described is set out below: 



(1) 


(2) 


(3) 




(4) 


(5) 




0.000000 


4.912870 


4.870246 




.216884 


.000000 




9.829667 


4.870246 


4.809984 




.216884 


9.726781- 


-10 


9.659333 


4.809984 


4.693208 




.216884 


9.379409- 


-10 


9.489000 


4.693208 


4.418914 




.216884 


8.818006- 


-10 


9.318666 


4.418914 


3.739177 




.216884 


7.693641- 


-10 


9.148333 


3.739177 


.477121 




.216884 


3.581515- 


-10 


8.978000 


0.477121 


.000000 




.216884 


9.667005- 


-10 




(6) 


A 




A2 


A3 




(0) 


1.000000 


- .466934 


+ 


. 173425 


- .053706 




(10) 


.533066 


- .293509 


+ 


.119719 






(20) 


.239557 


-" .173790 










(30) 


.065767 


- 










(40) 


.004939 












(50) 


.000001 












(60) 


.000000 


X 10 






18.433300 






1.843330 




-n + 1 














XI 

2 




= -5.500000 






n ! -l 














X- 


.466934 


- - .385221 







12n 



■n 2 -l 



X + .173425 



24n 
(n 2 -l) (19n 2 -l) 



720n 



X-. 053706 
o35:45 



.071538 



014017 



5.970776 



12.462524 



Having now the joint life annuity on the two lives, for the single 
premium according to the formula selected, we have (13 . 462524-^- 1.04) 
— 12 . 462524 = . 4822, and for the annual premium by the familiar 
formula, we have : . 48224- 13 . 4625 = . 03582. The latter sum multi- 
plied by 1000, gives the net annual premium for 1000, i. e., 35.82. 



CONTINGENT AND SURVIVORSHIP INSURANCES. 201 

107. Columns 1, 2 and 3 above will be recognized as v n , and the 
logs of lx+n and l y + n , respectively, and the effect of adding their 
logs is to take their product, that is, to multiply them together. Adding 
on the co-log of l x and l y is equivalent to dividing the products of 
1, 2 and 3 by the product of l x and l y . The work may therefore be 
performed, but with more labor, arithmetically and without resorting 
to logarithms. By means of the process just illustrated, the premiums 
may be readity computed with no other data than a life table and a 
table of v n . 

108. The insurance considered in the last article pays 1000 to the 
survivor at the end of the year of the death of either A or B. Had the 
policy provided that the benefit should be paid only to B, that is, that 
it should be paid only upon the contingency that B should survive A, 
it would be called Contingent Insurance or Survivorship Insurance. 
The symbol assigned it is, A xy . 

109. To compute such insurance without previously prepared 
commutation tables requires considerable labor. This book affords 
facilities for computing joint life insurances at the rate of 372% on 
the American Experience and at 4% on the Actuaries table, on two 
and three lives, by means of the joint life equal age Commutation and 
Annuity tables, used in connection with the force of mortality tables 
in the manner explained in connection with life annuities (Articles 
36-39, Chapt. IX). By means of formulas already given contingent 
insurances may be readily derived from the joint insurances. 

110. Let it first be required to find the single and annual premiums 
of a joint life insurance of $1000 on the lives of A, aged 28, and B, aged 
40, by the American Experience table at 372%. 

M columns have not been furnished, but we may compute the 
insurance from the 372% annuity table. First, we find the force of 
mortality, u. 28 = .00821, [x 4 o = .00977, the average is .00899. That is, 
greater than the force of age 35 which is .00888 and below 36 which 
is .00902. The difference of the last two is .00014, and the difference 
between the average force and the last is .00003. The difference in 
the joint fife annuities at the same ages and rates is by the table, 
14 . 57817 - 14 . 36078 = . 21739. 3 / 14 of the latter sum is . 04667, which 
must be added to the annuity for age 36:36, and we have the annuity 
14 . 40745. Adding one to make the annuity immediate, then dividing 
by 1 .035 and deducting 14.40745 according to the formula 51, we have 
for the single premium, . 48284 for 1 . 00 or $482 . 84 for a policy of $1000 
And by formula 53, A xy 

P xy = , and we have .48284-M5.40745, 

1 + Ox y 

.031344 or $31 .34, as the annual premium for a policy of $1000.00. 

110. Now let the contract be such that the insurance shall be 
payable only to B and only in case he shall survive A. What then 
would be the premium? 



202 FINANCE AND LIFE INSURANCE. 

111. For the latter problem, the following formula has been 
deduced : % 



1 / Ox-x'.y o*:y-i\ 
. xy = I A xy H — I. 

2 \ Px-i p y -i / 



The value of p 2 7 and p S 9 we get from the mortality table (Table No. X) 
under the head "Yearly probability of living," and are respectively 
.99180 and .99041. The annuities for ages 27:40 and 28:39 may be 
computed from the equal age annuity table as illustrated in article 
109, and are respectively 14.57817 and 14.45416. The quotients are 
14 . 69782 and 14 . 59944. Inserting these values in the formula with the 
value of A 2 8-4o and solving and then dividing the result by two, we have 
$290.61 as the required single premium. The annual premium would 
be found by dividing by 1 +a xy as before. 

112. To find the value of a contingent insurance payable to A in 
case he shall survive B, we have the formula: 

A X y == A !C y A X y. (57) 

It follows that the single premium in this case would be $482.84 — 
$290.61 =$192.23. The value may be found by a slight modification 
of the formula of Article 110. in that x and y are transposed so that 
the formula becomes: 



I=/a„ 



Py-i 




In the last two articles, the annuities were taken at the nearest approxi- 
mate equal age without interpolating for fractional parts of the years 
and in that respect, are only approximately correct. 

113. Suppose an estate is left to an institution contingent upon the 
death of both A and B, the death of B preceding that of A. That is 
it goes to B if A dies first, but goes to the institution on the death of 
A if he survives B. What is the value of the interest of the institution 
assuming the estate is valued at $100000? The value is that of an 
insurance on the life of A, less the value of an insurance contingent 
on his dying before B. Its symbol is: 

A xy =Ax-Ai y (58) 

114. To solve the problem of Article 113, we require the single 
premium on the life of A, aged 28. It is .32512. In Article 111, we 
computed an approximate value of A xy . It is .29062. And by for- 
mula 58, A xy = . 32512 - . 29062 = . 03450. Multiplying by 100000, we 
have as the present value of the interest sought, $3450.00. 

115. The force of mortality and regraduated annuity tables may 
be used to compute contingent insurances on two and three lives 
directly from the joint life annuity tables. We will propose three 
problems and state and apply the formulas for their solution as a con- 



CONTINGENT INSURANCES, FORMULAS AND SOLUTIONS. 203 

elusion to this chapter. Three lives will be considered, A aged 20, 
B aged 30, and C aged 40. The American Experience table of An- 
nuities, No. 28, at 6% will be employed. 

116. First, what will it cost A to buy an insurance on the life of 
B for $25000.00, contingent upon A being alive at the death of B to 
receive it? 

This is a survivorship insurance A xy and may be solved by the 
formula : 

A*i=Maxy + 1 /2)+72(ax:y-i-ax:y +1 ) (59) 

(x 3 o = .00835 and c xy = 11.973. Adding 50c it becomes 12.473. 020:29 
computed by the equal age method is 12.004 and 020:31 by the same 
method is 11.895, and we have A*£ = .00835 X 12. 473 + 7*(12. 004- 
11.895) = .15865. Multiplying by 25000, we get $3966.25. If we 
substitute z for y in the last problem, we would have A xz = • 00977 X 
11. 689 + 72(H. 286 -11. 082) = .21519. The insurance is payable 
immediately after death. 

117. Second, What is the value of an insurance payable to A upon 
the death of B, provided it occurs first of the three lives, A, B and C? 
For this value, the following formula has been demonstrated: 

A X y Z =^ya xyz + 7 2 + 72(ax:y-i:z-Ox:y + i: z ) (60) 

u, y = . 00835, a xys we have found, is 10.093. Adding 50c, we have 
10.593. We may compute 020:29:40 and 020:31:40 by the equal age method 
discussed in Chapter IX, and we find them to be 10.120 and 10.065, 
respectively. We now have A xyz = .00835x10.593 + 72(10. 120- 
10.065) = . 11595. Multiplying by 25000, we have $2898.75. If we 
substitute z in the place of y in the last problem, we will have A xy \ = 
00977X10.593 + 7 2 (10.081-10.010) = .13919. 

118. Finally, let it be required to find the present value of the 
insurance payable to A upon the contingency that he shall survive 
both B and C. 

For this value, we may use the symbol and formula: 

A.x'.yt = Axz+A x y A x ya A xya . (61) 

3 

We may employ the values found in the two preceding articles and we 

thus have: 

A*:~ = . 21519 + .15865 -.15195 -.13919 = .11870. 

3 

This multiplied by 25000, will give the value of the reversion or 
insurance to A . It is $2965 . 00. 

119. A formula for finding the single premium for an insurance 
of the form given in the last article is demonstrated in the Text Book 
of the Institute and is as follows : 

A xy : z =t JL x(a Z a — axy*+f.y(o ya — o xyz ) 

3 
+ 72 (o x -i : t ~ a x +i : > + o y -i : , — o y + i : ,) 

— 1 h{<hri'y-l'a — Ox+lly+l's) 

Contingent insurances involving three fives are not of frequent occur- 
rence and no more space need be devoted to them here. 



CHAPTER XI 

Of Valuation 

Insurance policies as a rule are based upon values derived from 
some reliable mortality table and computed at a rate of interest which 
it may be safely expected will be realized from the investment of in- 
surance funds. In this country, 3V2 per cent is now the most usual 
rate, while it is not unusual for companies to earn 4 x /2, or 5 or even 
6 per cent. This leaves a surplus unless taken up by expenses, un- 
favorable mortality or unfortunate investments. Upon policies so 
based, in case none of the latter conditions arise, there might be little 
occasion for frequent valuations so far as the interests of the companies 
are concerned. But the laws of some of the states require that annual 
distributions of surplus shall be made, and in many cases the com- 
panies issue policies contracting to pay annual dividends. Again, the 
laws of many, perhaps most, of the states, require companies doing 
business in the state to make reports to the insurance department 
showing solvency as a condition precedent to beginning or continuing 
to do business in the state. These circumstances require the valuation 
of the contracts of the companies outstanding for the purpose of ascer- 
taining the liabilities of the companies arising on the obligations of 
said contracts in order that these liabilities may be compared with 
the resources of the company and its solvency ascertained. The 
process of ascertaining this liability is called valuation. > The liability 
of an insurance company arising upon its contracts has strangely been 
assigned the name, reserve, and this designation taken in connection 
with one of the methods of net valuation sometimes employed, has 
often been misleading and in some instances, has led the courts into 
an erroneous and unscientific statement of the laws relating to valua- 
tion. The liability disclosed by a valuation of the outstanding con- 
tracts of an insurance company is not a reserve in any sense other 
than the liabilities of a commercial house upon its accounts or bills 
payable as disclosed by a balance sheet is a reserve. The word, 
however, is thoroughly inbedded in the nomenclature of the Actuarial 
profession and when properly understood, will answer as well as another. 
The text writers on the subject, as a rule, are careful enough to say that 
the reserve represents the sum which the company ought to have on 
hands as an offset to the obligations arising upon its policies. The 
Text Book of the Institute uses this form to which there can be no 
objection, when properly understood. 

"The value of the policy is thus seen to be the sum which the 
Assurance Office must have in hand to provide for that portion of the 
liability under the contract which the future premiums will not cover, 
and the only source from which it can be derived is the accumulations 
of the balance of past premiums not absorbed by the risk already 
incurred." 



VALUATION, PROSPECTIVE METHODS. 205 

In some instances, the last clause has been erroneously interpreted 
to mean that unless the sum required has been collected or accumulated 
from the balance of past premiums not absorbed, the liability (reserve) 
does not exist, a palpable non sequitur. Occasion for valuation also 
frequently arises on settlements on policies granting surrender values 
and also on lapsed policies under non-forfeiture laws, in force in several 
states. The theory of valuation will be readily understood from exam- 
ples and explanations which will now be given. 

1. As a standard for valuation, let us adopt for the present the 
Actuaries table and four per cent interest. Also as an example, let 
us take a person aged 30 years and $1000.00 as the amount insured, 
the valuation to be made five years after the date of the policy. For 
an ordinary life policy what is the net value? The symbol of value 
is V or nV, the n indicating the age of the policy at the time of val- 
uation. 

2. At the inception of the contract, as we have seen, the value 
of the annual net premiums and the value of the benefit exactly balance. 
That is, the net single premium is the mathematical equivalent of a 
life annuity of the annual net premiums. That is, the net single pre- 
mium, $306.17, is of the same value as a life annuity of $16.97, the 
net annual premium. At age 35, the value of the insurance has in- 
creased to $340.60 the policy now being five years nearer maturity 
but the annual premiums to be received are now worth only $290 . 93, 
because it also has approached nearer to its end. The value of a life 
annuity due of 1 at age 35 is 17 . 144 and the value of an annuity due of 
16 . 97 is 16 . 97 X 17 . 144 = 290 . 93. It now appears that the value of 
two sides of the policy no longer balance but the company will in the 
future receive $49.67 less than the present value of the benefit it is 
obligated by the contract to provide. The latter sum is the net 
value of the policy. From this, we derive the formula: 

nVx=A x + o-P(l+ffx + n) (1) 

3. We may suppose that our subject at age 35 will apply to an 
insurance company for another insurance in the same amount issued 
at his then age. The net annual premium would then be 19.87 or 
2 . 90 more than it was when he first applied 5 years earlier. Then the 
net annual premium, as we have seen, was 16 . 97. The value of 2 . 90 
per year for life at age 35 is that of an annuity of that sum or 2 . 90 X 
17. 144 =49. 72. This is the amount of the extra liability on the first 
policy because of the difference in the premiums. The formula is 
therefore 

nV x = (P x + n -P x ) 1+tfx + n (2) 

4. By substituting in formula 1, the values of A x+n and P x in terms 
of annuities and transforming the result algebraically, a formula is 
developed for finding nV x from the annuities alone. This formula is 
usually stated as follows: 

Ox-Oi + n 

nV s = (3) 

1+Ox 



206 FINANCE AND LIFE INSURANCE. 

5. Solving the problem of Article 1 by formula 3, we have the 
following factors and result: 

[ (17. 040 - 16. 144) -j- 18. 04] X 1000 =49. 72. 

6. When only a table of net annual premiums is at hand, the 
following formula may be employed. 

-tx + n i x 

nV x = (4) 

Px+n+d 

7. Solving our problem by this formula, we have the following 
factors and result: 

[(.01987 -.01697) +(.01987 + . 03846)] X1000 =49. 72. 

8. The value may be found, using a table of single premiums 
only by the following formula: 

Ax+n -A- X 

nV x = (5) 

1-A X 

9. Solving the same problem, taking the single premium for 1 
at the two ages the figures are as follows: 

[(.34060 -.30617) -HI - .30617)] X 1000 =49. 63. 

Note: The five preceding formulas are examples of what is 
termed the prospective method of valuation. 

10. If it be assumed that net annual premiums fixed at age 30 
have been exacted and accumulated during the years from age 30 to 
35 the net value may be computed by a method which well illustrates 
some of the principles of life insurance. Having the net annual 
premium, 16.97, paid down in advance the first. year we place it at 
interest at 4 per cent and at the end of the year we have 16 . 97 X 1 . 04 = 
17.649. From the q x column of the fundamental table at age 30, 
we get the figures .008425, which multiplied by 1000, gives us 8.425 
as the cost of the insurance for one year on the life of our subject. 
This deducted from 17.649 gives 9.224 as the terminal net value 
at the end of the first year. This 9 . 22 is now available to pay death 
losses and only 1000—9.22=990.78 is now to be insured. We find 
qn to be .008578. This multiplied by 990.78 gives 8.49 as the cost 
of the insurance the second year. Add 9 . 22 to the annual premium 
16.97 and increase it by 4%, making 27.24, from which deduct cost 
as above found and we have 18 . 75 as the terminal net value at the end 
of the second year. This course pursued successively will give the 
net values for each year. The value of one being computed from 
the previous one. The work is much shortened when the costs have 
been worked out and tabulated. We may formulate an equation 
representing this process, using S to represent the sum insured and 
nV x the value of a policy issued at age x after n years. 

n+iV x = (l+i) (P+nVO-qx+n+i (S-nV x ) (6) 



VALUATION, FORMULAS AND EXAMPLES. 207 

11. On principles elucidated in Articles 94 and 95 of the preceding 
chapter, the following retrospective formula may be developed : 

P(N X -, -Nx+nij - (M x -M x+n ) 

nV x = (7) 

D x + n 

12. Solving our familiar problem by the last formula, it works 
out as follows: 

.01697 (479951. 73 -358785. 45) -(8145. 75 -7127. 86) 

— ■ = .04959. 

20927.30 

Multiplying by 1000, we obtain the net value 49.59, substantially as 
before. In this problem, we have assumed that both the premiums 
and the insurance costs have been forborne, that is allowed to accumulate 
for five years. The excess of the value of the premiums over the value 
of the accumulated annual costs is, of course, the value of the policy. 
By this process it is not necessary to have or compute the values of 
the intermediate years. 

13. Formula (7) may be reduced by canceling M x by P x (N x — i), 
which is its equivalent so that it may be stated in a simpler form, thus : 

M x+n -P x (N x+n - 1 ) 

nV x = (8) 

D x+n 

14. The work required by the last formula is shorter, requiring 
but three values from the table and the net annual premium. 

7127. 86 -.01697 =58785. 45-=- 20927. 30 = (7127. 86 -6088. 59 

1039. 27) -=-20927. =.049661, which gives 49.66 for the 1000 policy. 

15. In the case of limited premium whole life policies exactly 
the same principles govern. If we suppose the insurance we have been 
discussing is to be paid for by, say 20 annual premiums, we have seen 
that the annual premium is A x -M+a x n Ii | , in this ease, 306. 17-M3.087 

= 23 . 39. After five years, the insurance is worth 340 . 60, but the fifteen 
annual premiums which the company is still to receive are only worth 
23.39 into a 15-year annuity due at age 35, that is, 23.39X10.848 = 
253 . 72, which is 86 . 88 less than the then value of the insurance. 

16. The process employed in Article 15 may be expressed by the 
following formula representing the number of premiums contracted by r. 
nV x =A x+n -rP x (l + |r-n-la x+n ) (9) 

17. Formula 7 may be employed to solve the problem of Article 
15 merely substituting the annual premium for limited payments in 
place of P x . This is true because by that formula the annual premiums 
and mortality costs are simply accumulated and compared to find the 
value. Obviously, formula 8 can not be so employed because rP(N x -i) 
is not the equivalent of P x (N x -i), and hence, not of M x . 

18. The value of a limited payment policy may also be found by 
formula 2, since the value of the policy at any year subsequent to its 



208 FINANCE AND LIFE INSURANCE. 

date is the value of an annuity (in this case a term annuity) of the 
difference in the premiums for the initial and later ages. Thus rP x = 
23.39, r-v-nP i+n =31.40. The annual saving on the last 15 years is 
therefore 8.01. This sum into a 15-year annuity at age 35, that is 
10 . 848, gives us 86 . 89 as the policy value. 

19. We have seen that an endowment insurance combines a term 
insurance with an endowment payable in case the term is survived. 
Let us substitute a twenty-year endowment insurance in place of the 
contracts just considered. Its value at the beginning is 496 . 65. The 
annual premium payable during the term is 37 . 95. The value of the 
benefit at age 35 is 582.77. At that time there are 15 payments to 
be received, the present value of which is 37.95x10.848=411.68. 
The value of the benefit therefore exceeds the future premiums payable 
under the contract by 582 . 77 -411 . 68 = 171 . 04. 

20. The process of the last solution may be represented in a for- 
mula as follows: 

rVP^hAx+r^l-Px + nd + ln-r-xax+r) (10) 

(n =20, r =5 in the last problem). 

21. Formula 2 may likewise be adapted to the solution of the 
problem of Article 19, thus: 

rVP x ;|=(P x+ ~|-Pxn) l + |n-r- l0x+r ) (11) 

22. Let us now suppose that our subject has contracted a twenty- 
year temporary insurance. What is its value at the end of five years? 

Solution: The single premium for the insurance is 129.04 and is 
equivalent to an annual premium of 129.04-^-13.087=9.86. The 
single premium to purchase a 15-year term insurance at age 35 is 
115.31. But the value of 15 premiums of 9.86 at age 35, is 9.86 X 
10 . 848 = 106 . 96. The balance of the account then stands in favor of 
the insured, 115.31-106.96=8.35. 

23. From the foregoing, we deduce the formula: 

rV|nP x = |n-rA x+r -|nP x (l + |n-r-lax+ r (12) 

24. The problem of Article 22 may be solved by the method of 
formula 2, that is, by finding the present value of the difference in the 
net annual premiums at the initial age and the age when the valuation 
is made. 

25. For the computation of net values by a continuous retrospec- 
tive method, a formula was devised by Elizur Wright, a famous Ameri- 
can actuary, as follows: 

nTlVx=u x+n (nV x +P x -c i+n ) (13) 



VALUATION, REMARKS AND DEFINITIONS. 209 

26. For convenience in applying the foregoing formula, tables of 

1+i D x 
u x and c x have been computed u x being equivalent to = 

Px Dx+l 

and c x being equivalent to q x discounted one year, that is, . 

1+i. 

The first year the net value would be u(P x — c x ). The second year, 
lVx would be added to the premium and c x +i deducted and this result 
multiplied by u x +i and so on. (13) is the general expression of the 
formula. 

27. The problems considered in the foregoing articles relate to 
the simpler and more usual forms of insurance, but any policy whatever, 
having fixed premiums and fixed benefits, may be valued by comparing 
the value of the benefit promised with the value of the premiums, 
promised. Both values being computed by the same standard of mortal- 
ity and rate of interest. Conversely, no policy can be valued, however 
much may have been paid upon it in the past in the way of premiums if 
either the future premiums or the benefit is not fixed and certain. 
Like an account, both sides must be known before a balance can be 
struck. The valuation formulas considered all relate to net premiums. 
That is, the pure premiums derived from the table of mortality and 
rate of interest employed, without taking into consideration the ex- 
penses of transacting the business or profits to the insurer. It may be 
worth while to add that the processes of net valuation are purely 
mathematical and do not involve metaphysical or ethical considera- 
tions at all. There can no more be an "equitable theory" of net valua- 
tion, as has sometimes been supposed by courts dealing with the sub- 
ject, than there can be of the Binominal theorem or Taylor's Formula 

28. The remainder of this chapter will be devoted to the discussion 
of various problems involving questions of policy values, surrender 
values and non-forfeiture contracts, first for convenience giving a few 
definitions. 

29. , Net value, we already understand to be the difference between 
the present value at the date of valuation, of the benefit insured and 
the present value at the same date of the future net premiums indicated 
by the form of the policy. In net valuation net premiums are employed 
and the gross premium otherwise called the office or contract premium 
is not considered at all except to determine the form of the policy 
unless it should happen to be less than the net premium. If the gross 
premium indicates an ordinary life, limited pay life, term, step rate or 
endowment insurance, the net premiums will be computed on that 
form of policy, but the amount of the gross premium, if adequate, will 
not affect the valuation otherwise in any respect. 

30. Surrender Value is defined in Principles and Practice of 
Life Insurance as being the equivalent given by an insurance company 



210 NONFORFEITURE. PAID UP AND EXTENDED INSURANCE 

to a policy holder on giving up his rights under the policy. Such 
surrender values are usually payable in one of three forms: (1) Cash; 
(2) Paid-up Insurance ; or (3) Extended Insurance. 

31. Non-Forfeiture laws or contracts relate to statutory pro- 
visions or to clauses in insurance policies which provide that all or 
some portion of the net value shall not be forfeited to the company in 
case of the lapse of the policy for failure to pay premiums when due. 
It was long the custom of insurance companies to provide in their 
policies that all premiums paid should be forfeited in case of breach 
of any of the numerous conditions of the policy and by this means the 
liability of the company, which was in some cases large, was cancelled 
without consideration to the policy holder, because of his inability or 
unwillingness to perform some condition or make further payments. 
It was largely through the initiative of Elizur Wright, that the injustice 
of this practice was exposed and the legislature of Massachusetts 
induced to pass a law requiring the company to apply a portion of the 
net value of the policy to the purchase of extended insurance. Many 
of the states have adopted the principle of such legislation and either 
require the companies to incorporate such provisions in their policies 
as a condition on which they are permitted to do business in the state 
or else the provision is incorporated in the policy by the force of the 
statute itself. At the present time, the companies, as a general thing, 
contract for surrender values in some form, and in many instances 

these are more liberal than those required by statute. 

> 

32. Paid-up Insurance is usually understood to mean whole 
life insurance for which no further premiums are to be received. 

33. Extended Insurance contemplates the continuation of the 
insurance written in the policy for such a term as the surrender value 
applied as a single premium will purchase. 

34. A simple method of computing the amount of paid-up insurance 
which a single premium will purchase is to solve by proportion. Thus, 
suppose the policy of Article 15 is to be settled by applying three- 
fourths of the net value to the purchase of paid-up insurance. What 
would be the amount of paid-up insurance granted? The net value 
being 86.88, three-fourths would provide a net single premium of 
65.16. The net single premium for 1000 at age 35, is as we saw, 
340.60 and we have the proportion 340.60 : 65.16 :: 1000 : (x). 
Solving for x we have 191 .40 as the face of the paid-up policy. 

35. There are various methods of arriving at the term of extended 
insurance which a given net premium will purchase, one or two 
which may be used without tables other than those given in this book, 
will be elucidated. 

36. If the single premium is small, that is, if by inspection it 
appears that the term will be short, a practical method will be to 
improve the single premium at interest one year, then deduct the cost 
of one year's insurance which may be easily found by using the q x 



COMPUTING TERM OF EXTENDED INSURANCE 211 

table, then improve the balance and deduct as before until the whole 
is absorbed. In case a fractional premium remains for the last year, 
it wall purchase such a proportion of a year's insurance as the fraction 
bears to the cost of a full year's insurance for that year, it being the 
practice to assume that the death rate is uniform throughout the year. 
The difference between the age of valuation and the age at the end of 
the term is the term sought. 

M x - M x+n 

37. We have seen that ]nA x = . Then |nA x D x = 

D x 

Mx— M x+ ni 1 being assumed as the capital sum insured. If A x be 
changed to A' x one of two results must follow, either the sum insured 
must be changed or the term must be changed. Let us suppose that 
the term n is changed to n', an unknown quantity. We then have 
|n'A x D x =M x — Mx+n 7 . |n'Ax is known, being the given single 
premium. By transposing the terms of the last equation, we have the 
equation M x — D x (|nA x ) = M x +n' in which the unknown quantity 
Mx+n stands alone and we may now solve the problem. 

38. Let us take the single premium of Article 34 and apply it to 
purchase extended insurance. First we divide the premium 65 . 16 by 
1000 to reduce it to the basis of unity. Substituting this and the 
values D x and M x in the above equation, we have: 

7127. 83- (.06516x20927. 30) =(7127. 86- 1363. 62) =5764. 24 =M x+n '. 
Looking now at the table of M, we find 5764.77 opposite age 43 from 
which we learn that the term is 43—35 or eight years — approximately. 
It rarely happens that Mx+n' falls upon a complete year as in the above 
case. When it does not, Mx+n' is taken to the nearest complete year 
and we interpolate for the months and days beyond the complete 
years. 

39. As another example, let it be required to find the term of the 
extended insurance which the net value found in Article 2 on the life 
considered in that article will purchase. On an insurance of 1, the 
single premium would be . 04967. Employing this in the formula, we 
have : 7127 . 86- ( . 04967 X 20927 . 3) = 7127 . 86- 1039 . 46 = 6088 . 40. 
Looking down the M x column, we find 6079 . 19 as the largest number 
within the last number. This is opposite age 41. Taking the difference 
of these numbers, we have 9.21 excess to be considered. The next 
number above M 41 is 5920.13. Deducting M 42 from M«, we have 
for the whole year, 6079 . 19- 5920 . 13 = 159 . 06. Now, 159.06:9.21 ::365: 
(x days). Solving for x, we have 21 . 7 days which in practice is called 
22. The term of extended insurance is therefore 6 years 22 days. 

40. In Mr. Dawson's book, Practical Lessons in Actuarial Science, 
the time represented by the surplus after deducting M x +n' from M x+n 
(9.21 in the above example) is called w and for finding its value he 
gives the following formula: 

365 (M* +n -M x+n ') 
w = (14) 

M x + n — Mx + n+l 



212 NONFORFEITURE PROBLEM— SOLUTION 

41. Let it be required to find the net value and term of extended 
insurance of a policy for 3000 insured at age 35 on the 20-pay life 
plan, the annual premium being 113. 10, one-half the annual premiums 
to be returned in case of death during the premium-paying period, the 
valuation to be made at age 45 on Actuaries table and 4%. Let us 
employ the prospective method of valuation. The first thing required 
is the net annual premium. The benefit insured consists of two 
parts, the capital sum, 3000, and a temporary increasing insurance of 
56.55 per year for 20 years. The net annual premium of the first is 

M 35 
X3000= 7127.86 -5- (385785.34 -87924. 72) = .026315X3000 

N 3 4— N54 
= 78.95. 

For the second item the net annual premium is : 

R 3 5-R55-20(M 5 5) 



X56.55 



N34— N54 



167460 . 34- 56143 . 92- 20 X 3958 . 84^ 385785 . 34- 87924 . 72 = 
. 1 18695 X 56 . 55 = 6 . 712. The total net annual premium is the sum of 
78 . 95 and 6 . 71 =85 . 66. The two branches of the solution might have 
been combined as shown in treating of annual premiums by uniting the 
numerators, thus saving one division since both denominators are the 
same, but the solution may be better understood by the method 
pursued. We next are to find the value of the insurance at age 45 at 
which the valuation is to be made. The value of the capital sum insured 

M 45 

at age 45 is X3000 = 1285.71. The value of the insurance on 

D 45 

premiums consists of ten year term insurance for 565 . 50 arising on the 
ten premiums already paid and an increasing temporary insurance 
for the remaining ten premiums. Combining these, the single prem- 
iums for this part of the benefit may be expressed as follows : 

(M45- M 55 )565 . 50+ (R45- R 56 - 10(M 55 )56 . 55 

-„ _ 107 . 898. Combining 

D45 

the values found, the present value of the insurance is 1285 . 71+ 107 . 90 
= 1393 . 61. The present value of the future premiums is P(l+9a 4 5) = 
85 . 66 X 7 . 957 = 681 . 59. Deducting the value of the future premiums 
from the value of the insurance, we have as the net value of the policy, 
712 . 02. If the*3000 only is to be carried forward as extended insurance 
the term would be found as in Articles 38 and 39 and the problem 
presents no difficulties to one familiar with those articles. The prob- 
lem, however, contemplates the return of one-half the gross premiums 
paid in case of death during the premium paying period. It would be 
proper then to carry the 565 . 50 along with the 3000 for ten years and 
continue the_3000 thereafter until the single premium is exhausted. A 



MEASURE OF DAMAGES, ON CANCELLED POLICY 213 

simple plan to pursue in case it is apparent that the term will extend 

be3^ond the term of the temporary insurance would be to find at once 

the present value of the term insurance and deduct it from the single 

premium and apply the balance to extend the face of the policy. 

Proceeding in this way, we have: 

M 45 — M 55 

X 565. 50 = (5461 .36- 3958. 84) -r- 12743. 15 = . 11791 X 565.50 

= 67.68. Deducting this sum from 712.02, we have as a single 
premium for 3000 term insurance, 644 . 34 or . 21478 for an insurance of 1 . 
D, 5 X .21478 =2737.23 and 5461 .36 =M 45 -2737.23 =2724. 13. 
M 6 s =2722.87 and the term is approximately 63—45=18 years. 
Solving for the parts of a year according to formula 14 gives three 
days, nearly, additional, so that the surrender value of the policy is 
3565 . 50 insurance for ten years and 3000 insurance for 8 years, 3 days, 
additional. 

42. Let us suppose that a 20-pay life policy issued at age 35 for 
1000 is wrongfully cancelled by the company after ten years, what 
value would be a proper measure of damages? 

The courts have taken different views of this question and without 
undertaking to determine the proper one, a few answers will be sug- 
gested, on the supposition that the object of the law is to compensate 
the policy-holder for his pecuniary loss. 

1. If only the facts stated in the problem are submitted, the net 
value of the policy computed upon the mortality table and at the rate 
of interest prescribed for valuations in the state where the cause arose 
would be a proper measure. If the standard of valuation were American 
Experience and 37*2%, this value, computed by formula (9), would 
be 291.42. 

43. 2. If it were admitted or proven that the office premium of 
regular and solvent insurance companies as generally charged, is, say, 
38 . 35 for a 20-pay life policy at age 35 and 74 . 79 for a ten-payment life 
policy at age 45 and that the policy-holder was still insurable and 
desired insurance paid up at age 55, a fair compensation, and perhaps 
the most fair, that could be awarded would be to allow him the present 
value of the difference in the two premiums, that is, (74.79—38.35) 
1+ |9a 45 , or 36 . 44 X8 . 17 =.$297 . 71. The reason why the latter value 
is regarded as proper is that policy-holders do not do business in terms 
of net premiums. On the other hand gross premiums in reliable com- 
panies do not materially differ for the same risks and such premiums 
bear great analogy to the market value of the commodities dealt with 
in the business world, and indeed are less variable than the value of 
most commodities. The market value is an accepted standard in 
measuring damages in the commercial world and is a fair criterion in 
the case supposed in this Article. 

44. Suppose the policy-holder by reason of age or infirmity is no 
longer insurable, what is the measure of his damages? 



214 MEASURE OF DAMAGES FOR BREACH OF INSURANCE CONTRACT 

The net value of the policy, of course, would be the minimum, 
for the reason that that amount of liability is cancelled 'on the com- 
pany's books by the termination of the policy and the company could 
not expect to profit by its fault. If the physical condition of the in- 
sured should be proven with certainty to be such that his life could not 
be prolonged beyond a certain brief period, it is possible that a larger 
recovery would be permitted. For instance, if the insured was 
afflicted with a cancer, tuberculosis or some other disease upon which 
medical witnesses could base a positive opinion fixing a maximum 
brief term of life, the damages measured by the interests of the bene- 
ficiary as well as relief upon the resources of the company, would justify 
a larger compensation, corresponding to the present value of an endow- 
ment insurance for the maximum period of the life, reduced of course, 
by the present value of premiums which would accrue during the term. 
It might be shown that the risk might be insured as an impaired life 
at a higher rate, in which case, the damages would be the present value 
of the difference in the two rates, computed on the principle of for- 
mula (2). 

45. If the insurance did not attach and was not in force from the 
date of the policy, the total premiums paid, with interest at the legal 
rate in force in the jurisdiction of the contract, would be a proper 
measure of damages, and cases may be easily supposed where further 
damages might be sustained growing out of the failure of the company 
to furnish insurance for the future as contemplated in the original 
negotiations, but it is improper, at least from an actuarial standpoint, 
to allow the recovery of back premiums when the risk has attached 
and the policy was in force, though this has been permitted in a few 
instances. 

46. The retrospective method of computing the damages, that 
is, the method of improving the annual premiums at interest and de- 
ducting the mortality cost, is wholly unsuitable as a means of ascer- 
taining the damages in any case unless net premiums are used in the 
computation or unless the tabular mortality cost and expenses are 
both deducted, it being supposed that the company has earned both 
the expenses and the current mortality costs while the policy was in 
force. 

47. The expectation of life treated as a term certain either for 
life insurance or for annuities is not a proper criterion for use in any 
case in estimating damages. Its employment for that purpose is 
unscientific and leads to incorrect results. 

48. In case dividends or bonuses of fixed amount are guaranteed 
by the contract, they constitute a part of the benefit insured and are 
to be considered along with the face of the policy in valuing the policy. 
When these dividends are made payable annually and vary in amount 
as they frequently do, they present some difficulty because of the 
arithmetical labor involved in the calculations. When the value of 
the dividends is represented in an agreement to increase the benefit at 
the end of the premium paying period by a sum certain in consideration 
of the forbearance and cancellation of the dividends ; or, to pay a gross 



VALUATION OF SPECIAL FORMS OF POLICY 215 

sum in cash in discharge of them ; or, when the premium-paying period 
is shortened because of forbearance to collect such dividends, the work 
is much simplified. 

49. Suppose, for example, a 20-payment life policy, issued at age 
35 for $1000, with guaranteed annual dividends during the premium- 
paying period, with the option, to accept in lieu of dividends the sum 
of $260.00 in cash, 459.00 in increased paid-up insurance at the end 
of 20 years, or that the premium paying period should be reduced to 
16 years, if the dividends were allowed to accumulate during that 
period, so that the policy would become paid up for $1000 in 16 years. 
In this case, we may regard the 260 as an endowment payable in 20 
years. The formula for a pure endowment we have seen is: 

D s4n 

nE x = . In this case, employing American Experience and S 1 / 2 % 

D x 
the figures are (9733 . 40-=- 24544 . 70) X260 = 103 . 11. The latter sum 
represents the present value of the annual dividends as reflected in 
260 cash settlement offered for them in the policy and they for the 
purposes of valuation may be treated as single premium for insurance. 
Now the single premium for 1000 insurance at age 35, is 370.55. Adding 
103 . 1 1 the new single premium 473 . 66 would- represent a policy for 
1278.20. This policy is the mathematical equivalent of the original 
policy and may be used as the basis for computing the net premiums, 
the net value and term of extended insurance upon said policy. 

Again, $459.00 paid up insurance at age 55 has a present value of 
259.86. The present value of this sum 20 years earlier is 260v 20 20p 35 
= $103.11 as before. From this, we may find the net single premium 
and equivalent whole life policy $1278.20, which may be valued ac- 
cording to formulas (1), (2), (3) or others. 

Finally, a policy paid up at age 35+16 or 51, is purchased by an 
annual premium of 370.55-i- 1-f- a 15 | and on the theory that the 
amount of the gross or office premium, if adequate, does not affect 
the net value of the policy, the policy supposed may be valued directly 
as a 16-payment whole life policy. 

50. Let us now consider the effect on the value of a policy, which 
will arise in case the gross or office premium is less than the net annual 
premium. The process of valuation as already shown is intended to 
show the liability of the company upon its contract. This liability 
we have seen, may always be found by equating against the value of 
the insurance promised in the policy, the value of the future premiums 
to be received by the company under the contract. Let us suppose 
that the gross annual premium of the policy of the last previous article 
was $25.00, what would be the net value of the policy at the end of 
ten years? 

The present value of the benefit, $1000, payable at the death of a 
person aged 35, by the American Experience table and S l /t% is $370 .55. 
The net annual premium for this benefit is $27.40. That is a twenty- 



216 TERMINAL AND MEAN RESERVES 

year annuity due of $27.40 is the mathematical equivalent of $370.55 
in cash. But by the conditions of the contract the company, while 
assuming a liability of $370.55, has done so for a consideration of 
$2 . 40 less each year than a sum sufficient to offset this liability. The 
value of this annual shortage is (l+|n— la x ) 2.40 or 13.889X2.40 = 
33.334. So that the policy is worth $33.33 more than cost at its 
inception. The company by its contract, has assumed this liability. 
At the end of ten years, the period of valuation supposed, the value of 
the benefit has increased to $456.00 and against this cash liability, 
the company has as an offset a ten-year annuity of $25 . 00, the present 
value of which is at age forty-five, 25 X8 . 170 = $204 . 52. The liability 
of the company therefore, is now $456.00—204.25=251.75, and the 
net value of the policy is therefore $251 . 75. Had the gross or contract 
premium been in excess of the net annual premium, the net value of 
the policy at the end of the tenth year would have been $232 . 19 or 
$19.56 less. In order, therefore to compute the net value of a policy 
having a gross premium which is less than the net premium it is neces- 
sary to add to the ordinary net value the present value of an annuity 
due of the difference between the net and the gross premium. 

51. The net value hereinbefore considered are what are known as 
terminal net values or terminal reserves. These values are appro- 
priate to individual policies. In the case of valuations for insurance 
companies, for the purpose of ascertaining their financial condition 
it is more practical to group the policies with reference to age and date 
of issue and it is usual to assume that all of the policies issued during 
a given insurance year were issued at the middle of the year on the 
theory that the shortage in interest on the premiums on the later 
policies will be offset by the excess on the earlier policies. The values 
thus found are called mean reserves. Such reserves may be found 
approximately from the terminal reserves by dividing the sum of the 
reserve for the n — lth year, the net annual premium and the reserve 
for the nth year by two. This gives the mean reserve for the nth year. 
Thus, the mean reserve for the problem of Article 1 may be found as 
follows : 4V 30 = 39 . 06,P 3 o = 16 . 97 and 5V a0 = 49 . 63 and (39 . 06 + 16 . 97 + 
49 . 63 ) -f- 2 = 52 . 82. For the mean reserve for the 6th year 5V X and 6V X 
will be substituted. As a formula the process may be stated as follows: 

n^lV x +P x +nV x 

nV x = (15) 

2 

52. In actions brought under the Federal Employers' Liability 
Acts, for deaths or personal injuries, the probabilities of life of the 
injured person in case of injuries not resulting in death and of both 
the deceased at the time of the injury and of the beneficiary claimant 
in cases of injuries resulting in death, enter into the estimation of the 
damages, and the apportionment of the same among the dependents 
in case of death. The courts have so far not gone further into the 
question of life contingencies than to allow the introduction in evidence 
of the tables showing the expectation of life of the injured person and 



EXPECTATION OF LIFE, PERSONAL INJURY TRIALS 217 

in case of his death, the expectation of life of the beneficiary also. 
Probably any of the tables published in this book (Table No. XXXI) 
would be regarded as standard and competent for such purposes. 
Certainly the one recognized in the state of the forum by statute for 
insurance valuations or for the valuation of life estates would be re- 
garded as competent. The value of the shares of plural dependents, 
such as a widow and minor children in the damages found, presents a 
question which could be solved with accuracy on the principles discussed 
in the chapters on life contingencies and annuities; but such nicety of 
computation has not been undertaken in the cases so far decided. 
Indeed this could hardly be done by a jury and even the court would 
probably require the assistance of the testimony of Actuaries. It has 
therefore been considered sufficient to resort to approximations based 
upon the tables of the expectation of life. Familiarity with the prin- 
ciples illustrated in the foregoing pages will, nevertheless, greatly 
aid counsel in the intelligent presentation of such cases to courts and 
juries. 



CHAPTER XII 
Of Surplus and Dividends 

The annual premium of an insurance policy usually consists of the 
net annual premium as a basis to which is added a "loading" also 
sometimes called "Margin" intended to cover the expenses incident 
to the business of writing the insurance and managing the business 
and also for the profit of the stockholders, if it be a stock company. 
This loading is ordinarily a certain per cent of the net premium, but 
it sometimes includes also a constant quantity bearing no relation to 
the net premium. 

The net premium is usually based upon a rate of interest somewhat 
less than the company might reasonably expect to earn on the funds 
of the company. The gross premium should always be made large 
enough to meet all the conditions necessary to assure the payment 
of every claim arising on the contracts of the company. It often 
happens that owing to favorable mortality, good management and 
other causes, the earnings of the company on its investments and sav- 
ings on expenses, the company will have a sum on hands in excess of 
the requirements for reserve and expenses. This excess is called 
surplus. It is about the same thing that is called "profits" in 
some other financial institutions. In mutual companies this surplus 
equitably belongs to the policy holders. In stock companies, in the 
absence of some legal or contractual obligation to the contrary, it 
belongs to the stockholders. But in both classes of companies, it is 
usual to set apart at least a portion of this surplus, for distribution 
among policy holders as dividends on their policies. Policies which 
are entitled to share in such dividends are» called participating 
policies. Those not so entitled are called non-participating pol- 
icies. It was usual in earlier years to place the distribution period 
at some distant future period, say ten, fifteen or twenty years, a prac- 
tice which resulted in disappointment to the policy holder, so frequently, 
that by legislation in some cases and by force of public opinion and to 
meet competition in others, it has generally become the practice to 
declare annual dividends after the policy has been force a few, usually 
two years. These dividends are, in this country, usually apportioned 
according to what is known as the "Contribution Plan," recommended 
by Sheppard Homans, an eminent actuary who also has the distinction 
of being the author of the American Experience table of mortality. 
The principle upon which this plan is based, is that each policy should 
participate in the distribution of surplus in the same proportion which 
it has contributed to produce the surplus. The modus operandi of 
the plan roughly stated is to charge against the policy its proportion 
of the expenses of management, the reserve required on the policy 
at the time and its proportion of the mortality cost of insurance ac- 
cording to the experience of the company, and crediting the policy 



SURPLUS AND DIVIDENDS, DISTRIBUTION 210 

with the gross premiums paid under the policy and net interest earned. 
The balance if in favor of the policy is the surplus. If at the time of 
computing the dividend there is a reserve to the credit of the policy 
for the previous year, it should be added to the gross premium and 
the interest computed on the sum at the rate earned by the company 
after deducting expenses incident to the investment. There are 
other sources from which surplus may arise, such as rents, lapsed or 
forfeited policies and other miscellaneous sources. These funds are 
disposed of in different ways. Sometimes by using them to reduce 
the item "expense of management" or by apportioning them to the 
credit side of the account along with the "net interest earned" or other 
uses are made of it according to the ideas of those managing the com- 
pany. Other methods of distributing surplus, no doubt, are employed 
in practice, but the subject need not receive further attention here 
It may be remarked that until dividends are declared, the surplus is 
the property of the company and the policy holder has at most a mere 
inchoate right therein, but after they are declared, they, if not paid, 
may under some circumstances, be used as an offset against liabilities 
to the company or to prevent forfeitures. 






CHAPTER XIII 
Of Fraternal Insurance 

A few words upon the subject of fraternal or mutual benefit societies 
may be helpful. These societies have been an immeasurable source 
of good to thousands of homes in this country and they should be so 
fostered and improved as to increase their usefulness. Very few of 
them have been constructed upon the scientific principles applied in 
the preceding chapters and, perhaps, it would not be possible to so 
construct them without destroying to some extent their fraternal 
characteristics. There is one thing that should be kept in mind, 
however, by those promoting or managing such societies, seeking 
membership therein or legislating for their welfare. It is certain that 
insofar as the insurance feature of the society is concerned, there is 
no magic in the name "fraternal." A man holding a policy in an 
"old line" insurance company and a certificate of membership in 
a fraternal society for like sums and at the same time, holds contracts 
imposing indentically the same obligations upon the two institutions 
in case of his death. It takes the same amount of the same kind of 
money in the case of each. The money in both cases alike must be 
paid by the members. True, it is in the case of old line companies 
the death benefit is largely met by interest accumulations, but this is, 
after all, merely earnings of excess premiums advanced by the insured 
so that in the last analysis the insured pay the death losses, all of them. 
The company, whatever its character, pays none of them. No insti- 
tution has yet been organized which has attained the philanthropic 
character of paying death benefits from its own resources. The 
membership which does not deny itself to create funds to meet the 
growing demands arising from increasing age and "keeps its reserve 
in its pockets" must accumulate in some other way the interest neces- 
sary to meet the increasing mortality costs that are bound to come. 
If the members do not each faithfully shoulder the burden of paying 
his own way, there must in the end remain a group of the faithful 
brethren, who while most worthy of all, will leave widows who will 
find their certificates of membership worthless paper containing 
promises which are delusions. 

If those who enter a society which does not demand of its members 
that they shall pay every year the full cost of their insurance, under- 
stand the fact that those who live to pay the most premiums will 
lose. If they understand this and are willing to accept the fact of 
living as full compensation for paying for others and receiving nothing 
for themselves, well and good. They should be allowed to do it. If 
one is induced to enter such a society without appreciating this fact 
he has been subjected to a cruel fraud which length of days will certainly 
disclose. This country has been the greatest field for the promotion 
of fraternal societies that the world has seen. The sovereign lodges 



FRATERNAL INSURANCE 221 

have had state after state to add to "the juri diction" to take in new 
blood and keep down the average death rate but with all this, the 
inexorable law of human mortality has had its way and thousands 
of men who had supposed their lives were insured for the whole of 
life have found that their hopes were vain, their certificates worthless. 
This has so often happened that all who want to know should know 
it. The lesson then is, that a rate of contribution should be established 
from the first that will exact from each member a contribution which 
at least equals the cost of his insurance for the period covered by the 
payment. In this way, his increased age will not affect prejudicially 
the new member who enters ten, twenty or fifty years later, and the 
new member, therefore, will have no cause to shun his society. Such 
a rate may be made from the tables and on the principles discussed in 
the preceding chapters, but the wise course would of course be to have 
them prepared by a competent actuary. As a corrollary to this 
proposition, the man who seeks membership in any of the various 
forms of mutual benefit insurance societies should first ascertain that 
his company is exacting of each of its members each year the cost of 
his insurance. This is the only guarantee of permanence. If it is 
not so doing, he should keep out. Self respect should deter him from 
seeking to cheat his brethren, should he die early, and prudence, 
from being himself cheated should he be favored with long life. 



INDEX 

Pci£f6 

A, Symbol for Single Premium - 162 

Actuaries Table of Mortality, original - - 103 

Actuaries Table of Mortality, re-graduated - - 104 

Actuarial Society of America — Acknowledgment - 3 

Accumulated or forborne annuities - - - 37 

Accumulated or forborne annuities, Table of - 78-84 

Adequate Rates, Fraternal Insurance - - 220 

Age, as affecting premium rates - 220 

American Experience, Mortality Table - - 105 

Amount in Sinking Fund at given period - - 39 

Annuities certain, in full - , - _ 34-52 

Annuities certain, symbol and formula - - 34 

Annuities certain, Tables of - - _ 86-92 

Annuities certain, How computed 34 

Annuities certain, Relation to compound amount - 38 

Annuities certain, payable in fractional payments - 35-36 

Annuities Examples, formula .and explanations 34, 35, 38-52 

Annuities due, defined - 34 
Annuities computing, by use of formula for the nth term, example 14 

Annuities Nature of, discussed - 39 

a\ Symbol for annuity - - - 163 

Annuity which I will purchase - 38-39 

Approximate Formulas for premiums - - 18 

Approximate Formulas for annuity value - 14, 199 

Annual Premiums, rules and formulas for computing 179-204 
Annuities, Life, discussion of - 162, 164, 167-177 

and therein of 

Ordinary Life Annuities - - 167, 168, 174 

Temporary Life Annuities - - 167, 173 

Deferred Life Annuities - - 167, 173 

Deferred Temporary Life Annuities - - 168 

Complete Life Annuities - 169 

Life Annuities payable fractionally - - 169 

Continuous Annuities - 169 

Joint Life Annuities - 170, 171 

Contingent Life Annuities - - 171 — 177 

Benefit and Premium, Equation - 178, 181, 182, 205 

Benefit of Policy, includes guaranteed dividends and bonuses 214 

Bond Valuation, Rules, Examples and Solutions 29, 40, 52 

Tables for 62—100 

Bonus on Policy, valued as part of benefit - - 214 

Bonus, as affecting bond valuation (Example 2) - 48 

Building and Loan, payment to mature 

share in given time, how computed - - 46 

Building and Loan, Term required to mature 

share by given payments, how computed - 47 

Carlisle Mortality Table 101 

Carlisle Commutation Columns - 110, 114, 116-119 

Commutation Columns, how computed - - 163, 166 

Commutation Columns, tables of - - 108, 131, 140-155 

Compound Interest, discussed - 25-30 

Compound Interest, Rules for Computing - 25, 42-43 

Compound Interest, Tables of at various rates - 62-68 

Compound Interest, how computed 25 

Compound Interest, nominal and effective rates explained 27, 28 



INDEX 223 

Compound Interest, nominal and effective rates, table of - 28 
Compound Interest, Extension of tables of compound interest 32 

Contingent or Reversionary Annuities - - 171-177 

C, a value used in computing M and R Columns 164-165 
Continuous Annuities - - - - 170-171 
Contribution plan of distributing surplus - - 218 
Coefficients of Lubbocks' formula (Sec. 17) - - 19 
Cost of Insurance - 206-209 
Curtesy Interests, initiate and consummate - 174 

D, Symbol of a Commutation Column - - 165 
Dawson, Miles M., work quoted - 212 
Deferred Annuity, defined 36 
Deferred Life Annuity, how computed - - 167-168 
Deferred Insurance defined - 183 
Deferred Insurance, Methods of computing premiums for 183-184 
Discount, defined and illustrated - 30-32 
Discount, Table of at various rates - - 70-76 
Discount, Table of how computed 31 
Discount, force of, defined and illustrated - - 33 
Discount or premium as affecting value of bond - 45-52 
Division, by contracted method - 5 
Division, by use of logarithms - - - 9 
Dower Interests. Inchoate and Consummate - 174 
Damages, on Cancelled insurance, considered - 213-214 
Damages, under Federal Employers Liability Acts - 216 
Errata, on page 104 the heads, "Yearly probability of dying 

qx" and "Yearly probability of living px," 4th and 

5th columns should be transposed - - 104 
On page 134 the annuity at age 10, four per cent, top 

of 4th line should be 12.90393 - - 134 

Endowment defined - 189 

Endowment premium for, how computed - - 190 

Endowment Insurance, defined - 189 

Endowment Insurance Premiums, how computed 190-192 

Endowment Insurance, how valued - 208 

Effective and nominal rates of interest — table - 28 

Effective and nominal rates, methods of computing 28-29 

Employer's Liability Acts, measuring damages - 216-217 
Expectation of life, table of by Carlisle, 

American Experience, Actuaries and Northampton 

tables ----- 139 
Expectation, not used in computing annuities or insurances 214 

Extended Insurance, defined - 210 

Extended Insurance Term of, how computed - 210-213 

Equivalent equal ages from different ages - 172-173 
Equal age method of computing annuities and 

insurances - - - 170-176, 201-203 

Extension of Compound interest tables example - 32 

Finite Differences. Employed in calculations 12-20, 41, 42, 200 

Force of Interest and discount - 33 
Force of Mortality by Actuaries, American & Carlisle Tables 132 

Force of Mortality Employed in Calculations - 172, 201-203 

Fraternal Congress Table of Mortality - 106-107 

Fraternal Congress Table Commutation Columns - 154 

Fraternal Insurance, premiums for - - 187 

Fraternal Insurance, remarks on - 220-221 

Guaranteed Additions, valued as part of benefit - 214 

Guaranteed Additions, Example of Valuation of - 215 

Gross Premiums, defined 178 

Gross Premiums, How produced from net premium - 197 

Gross Premiums, less than net premium — effect on value 215 



224 INDEX 

Gross Premiums, returned, valued as part of benefit - 212 

Hardy, G. F. formula of stated and applied - * 41-42 

Hunter, Arthur, credited - 170 

Homans, Sheppard, Author "Contribution plan" • - 218 

Industrial Insurance - - -. - 187 

Increasing Annuities - - - - 165 

Increasing insurances, computing premiums for - 194-198 
Increasing insurances, valuation of - - 212, 213 

Installment Policies, considered - - - 189 

Installment Loans, valuation of, examples - 47-51 

Intermediate terms of series, inserted by interpolation 14 

Interpolation, Methods of 13-15, 51 

Interpolation, examples illustrating - 7, 13, 15, 43, 51 

Interest rates, unusual, amount found by interpolation 42 

Interest, compound, rule for computing - - 26, 42 

Interest, compound, computing by use of logarithms 43 

Interest, compound, finding and computing at nominal rate, 

example _____ 43 

Interest, compound, convertible at fractional periods of year 35, 37 

Incomes, valuation and partition of, - - 171-177 

Same, examples and solutions - - 174-177 

Joint Life Annuities - - - - 170-177 

Joint Life Annuities, Equal age tables - 116-131, 134-138 

and therein are given commutation tables: — 

Carlisle's Two and Three lives, 5% \ - - 116-117 

Carlisle's Two and Three Lives, 6% - 118-119 

American Experience two and three lives 3 1-2% 120-121 

Actuaries, two lives 4% - 126 

Same, three lives 4% - - - 127 

Same, two lives 5% _,.-.,_ - - 128 

Same, three lives 5% 129 

Same, two lives 6% - - - - 130 

Same, three lives 6% ■ : -. - - 131 

And annuity tables: 

American Experience, two and three lives at 
3 1-2%, 5 and 6% 134-135 

Actuaries, two and three lives at 4, 5 and 6% 136-137 

Carlisle Experience two and three lives at 5 and 6% 138 
King, George, credited - - - 28, 41, 170 

1 and lx symbols, meaning living, those living at age x 22 

Landis, Abb, credited - 3, 189, 193 

Lagrange's Theorem, stated and illustrated - 16-18 

Loans, Problems relating to and Solutions, - 29, 42-52 

Logarithms, discussed - - - - . 6_11 

Logarithms, Short Tables of - - 56-61 

Logarithms, Mantissa and Characteristic - 6, 7, 8 

Logarithms, Operations in Multiplication, Division, Frac- 
tions, Involution, Evolution by 9 
Logarithms, Short table of, used as factors for computing 
logs and anti-logs to eight places of decimals ex- 
plained ----- 10-11 
Loaded premiums, a formula for - - 196, 197 
Loading or Margin, defined — purpose of - - 218 
Loading or Margin, not considered in net valuation - 209 
Lubbock's Formula, stated - - - - JJ 
Lubbock's Formula, Examples of computations by 19, 199-201 
Life Insurance, elementary principles of - 162-166, 184 
Makeham's Formula, remarks on - - 170, 172 
Mortality table, examples of graduation - - *% ^ 
Various tables published - - 101-107 
Multiplication by the contracted method - - 4 



INDEX 226 

Multiplication by the use of logarithms 9 

Municipal Bonds and Sinking Funds, various examples 45-46 

M, Symbol of a commutation column - - 165 

Measure of damages for breach of insurance contracts 213-214 
Monthly premiums, how computed - - 187-189 

Non-forfeiture, defined - 210 

Net value defined _____ 209 

Non-Participating policy, defined - 218 

Natural Premium Insurance, considered - - 187 

Northampton Mortality Table - -. - 102 

Northampton Commutation, N and D, Columns - 112-113 

Notation adopted and explained - - 23 — 24 

Notation — See various tables, formulae and explanations 
N, Symbol of a Commutation Column - - 165 

Nominal Interest Rates, table of - - - 28 

N'th Term, Formula for n'th term 13 

Same, use of illustrated - - 13, 15, 42 

Ordinary Life Policy, defined - 181 

Ordinary Life Policy, rules for computing premiums 165, 178-181 
Ordinary Life Policy rules for valuation of - 205-206 

Probability, discussion of subject - -. 21-24 

Probability, Mathematical probability defined - 21 

Probability, independent and dependent probabilities de- 
fined and illustrated 22 
Probabilities of human life discussed and illustrated - 23-24 
Probabilities of human life applied to life insurance and 

annuity problems - - - 162-166 

Practical Lessons in Actuarial Science, quoted - 212 

Principles and Practice of Life Insurance, quoted - 209 

Price which may be paid for bond to earn a given income 29-47 
Profits on Policies, treated as surplus - - 218 

Partition of Life Annuities - - - 171-177 

Premiums, Net Single and Office, distinguished - - 169 

Premiums, Single, defined - 169 

Premiums, Methods of computing - 163, 179, 201 

Premiums, Table of Single Premiums . - - 161 

Participating policies, defined - 218 

Paid up Insurance, defined - 210 

Paid up Insurance, how computed - - - 210-211 

Pension policies, considered - 189 

Policy payable instant of death, premium for - 97, 203 

p, px, symbols for probability of succeeding or living 22, 23, 164 
P. Symbol for net annual premium - - 179 

P', Symbol for Gross or Office premiums - - 196 

Principal of Loan, equivalent to V n - - - 30 

Principal of Loan, how found - 27 

q, qx, Symbols for probability of the failure of an event or 

the dying of life under observation - 22, 24, 164 

Rate of interest realized on instrument sold above or 

below par - 29, 49-52 

Rate of interest, continuous rate - 32-33 

Rate of interest, convertible fractionally - 28, 35-38 

Radix of Mortality Table - - - - 162 

R, Commutation columns explained - - 166, 194 

Reversion, equivalent to an insurance - - 177 

Reversionary Annuity, definition - 171 

Reversionary Annuity, Examples of 171, 177 

Remainder, valuation of, example - - 174, 202, 203 

Return Premium Policies, computing premiums for 194-197 

Return Premium Policies, Valuation of 212-213 

Reserves, defined - 204 



226 INDEX 

Reserves, Terminal, methods of computing - 205-212 

Reserves, Mean, defined - 216 

Reserves, Method of computing - 216 

Schedule of payments and interest of sinking fund 38-40 

Semi Endowment Insurance, premiums for - 191 

Series, Subject discussed - 12-20 

Series, Extending or supplying terms by finite differences 12, 13 

Series, Summation of by finite differences 18 

Series, Summation by Lubbock's formula - - - 18-20 

Series, development of, Methods ... 12-18 

Short Methods of Computation, discussed - - 1-5 

Special Premiums, symbols and formulas - 193-197 

Single Premiums, defined - 163 

Single Premiums, Methods of computing - 163, 178-195 

Single Premiums, table of 161 

Sinking Fund, defined and illustrated - - 38-39 

Sinking Fund, formulas for computing - - 38 

Sinking Fund, Schedule of payments and interest - 39 
Sinking Fund, when rate differs from rate borne by the 

loan-example - - - - 40 

Step Rate Premium defined and illustrated - - 193 

Step Rate Premium, Methods of computing - - 193 
Step Rate Premium, Example from Mr. Landis' book - 193 

Surplus, defined _____ 218 

Surplus, how distributed - 218-219 

Surrender Value defined - - - ' - 209 

S, Commutation Columns - - - - 165 

Terminal Reserves, Methods of computing - 205-216 

Term Annuities, how computed - 167-173 
Term Insurances, how computed - - 181, 182 
Term of loan, n, how computed - - 25, 30, 43, 46 
Time in which money at interest will double or otherwise 

multiply itself - - - - 30, 43 

Tontine Funds, defined, Formula - - . - 198 
Text Book of the Institute of Actuaries quoted 13, 15, 203, 205 

Theory of Finance, table adapted form - 28 

Tables published - - - 56-161 
and herein are given 

la. Logarithms to 8 places 1.0 to 9.99 - 56-59 

lb. Logarithms to 8 places 1.0000 to 1.00999 - 60-61 

II. Compound Amounts 1-4% to 10% - 62-68 

III. Present Values of 1 at 1-4 to 10% - 70-76 

IV. Amount of Annuity of 1 at 1-4 to 10% - 78-84 

V. Annuities Certain at 1-4 to 10% - - 86-92 

VI. Sinking Funds at 1-4 to 10% - - 94-100 

VII. The Carlisle Mortality Table - - 101 

VIII. The Northampton Mortality Table - 102 

IX. The Actuaries' Mortality Table - - 103 

The Actuaries Re-Graduated Table - 104 

X. The American Experience Mortality Table - 105 

XI. The Fraternal Congress Mortality Table 106-107 

XII. Actuaries N and D columns at 5 and 6% 108-109 

XIII. Carlisle N and D columns at 5 and 6% 110-111 

XIV. Northampton N and D columns at 5 and 6% 112-113 

XV. American Experience N and D columns at 5 

and 6% 114-115 

XVI. Carlisle Joint Life N and D columns at 5% 116-117 
XVIII. Same at 6% - - - 118-119 

XVIII. American Experience Joint Life N and D 
columns at 3 1-2% - - - 120-121 

XIX. Same at 5% - - - 122-123 



INDEX 227 

XX. Same at 6% - - - - 124-125 

XXI. Actuaries Joint Life N and D Columns, Two 

lives at 4% - - - - 126 

XXII. Same Three Lives at 4% - - 127 

XXIII. Same Two lives at 5% - - 128 

XXIV. Same Three Lives at 5% - - 129 

XXV. Same Two Lives at 6% - - 130 

XXVI. Same Three Lives at 6% - - 131 

XXVII. Force of Mortality by Actuaries, American 

and Carlisle Experience __- 132 

XXVIII. Joint Life Equal Age Annuities, American 
Experience - 134-135 

XXIX. Same Actuaries Experience - 136-137 

XXX. Same Carlisle Experience - - 138 

XXXI. Expectation of Life, Northampton, Carlisle 
Actuaries and American Experience - 139 

XXXII. Commutation Columns, Actuaries 3% 140-141 

XXXIII. Same 3 1-2% - - 142-143 

XXXIV. Same 4% - - 144-145 

XXXV. Same, American Experience 3% - 146-147 

XXXVI. Same 3 1-2% - - 148-149 

XXXVII. Same 4% - - 150-151 

XXXVIII. Same 4 1-2% - - 152-153 

XXXIX. National Fraternal Congress Commutation 
Columns 4% - - 154-155 

XL. Life Annuities, Actuaries 3 to 6% - 156 
XLI. Same, American Experience 3 to 6% - 158-159 
XLII. Same Carlisle Experience 4 to 6% - 160 
XLIII. Net Single premiums, Actuaries and Ameri- 
can Experience 3 to 4 1-2% - - 161 
Valuation of Investments 29, 44-52 
Value or present worth, v n , formula for 31 
Same, Tables of 70-76 
Value, Surrender value defined - - 209-210 
Varying Benefit in insurances - 194-198 
Varying Premiums in insurances - - 192, 193 
Value of Policy, a liability - 205, 215 
Valuation of Insurances, - 204-217 
and therein of the valuation of 

Ordinary Life Policies - 205-207 

Limited Pay Life Policies - - 208-209 

Temporary Insurances - 211-213 

Return Premium Policies - - 212, 213 

Policies with bonuses or other benefits - 215-216 

Mean Reserves - 215 

Terminal Reserves - 216 

Endowment Insurances - 208 
Policies with gross premium less than net premium 215 

Theory and purpose of - 205 

Processes of Mathematical - 209 

Wright's Continuous Method - - 208-209 

Wright Elizur, author of non-forfeiture laws - 210 

Wright Elizur, Valuation formulas - - 208, 209 

X, as symbol for age (See various tables) 23 



ERRATA 

Page 8. Art. 7. The illustration given in the last paragraph ' 
article is entirely erroneous, reversing the eor 
principle stated in Articles 3 and 4, same chap 
Page 8-9. Art. 9. The rule here given is correct but the reason n 

erroneously stated. Different powers of the s; 

factor and not different numbers, should be c 

sidered. 
Page 10. Art. 18. Insert the words "such number" after "and" in 

the 7th line. 
Page 14. Art. 8«. Substitute "a 3 o" for "A 3 o." 
Page 36. Art. 8. In last line substitute "448732" for "551267." 

Page 37. Art. 19. Substitute (l+7 P ) np -l 

in second member of 

m(l+ i / P ) P / m -l 
formula (23). 

Page 40. Art. 31. Substitute 3 per cent for 3 y% in example. 

Page 41. Art. 35. Substitute 'V " for "a" after the word annuity 
in first line. 

Page 43. Art. 44. Erase symbol, Sn| 

Page 44. Art. 46. Substitute the word "principal" for "rate" in 
third line of rule. 

Page 52. Table 6. Put "multiplying" for "dividing." 

Page 104. "px" and "qx" should be transposed and heads 

changed to correspond. 

Page 134. lOaxxx 5%, should be written 12.90393. 

Page 153. M 57 should be written 2611.373. 

Page 172. Art. 34. Change formula number to (17). 

Page 206. Art. 7. Change + between parentheses to -:- . 

In several places, the annuity symbol "-?." is 
improperly written "a" and "member" of equations 
improperly written "number." To avoid mistakes, 
note these errors at the places indicated. 



